\begin{appendices}
	\section{Poincar\'e inequality: local to global}\label{A}
		\begin{lemma}\label{p3.10}
	Let $x_0\in S$, $0<r_0\leq1$ and  $B=B(x_0,r_0)\subseteq S$,  $\rho_B=\frac{\rho\restr{B}}{\rho(B)}$.  Then the following strong local $(1,1)$-Poincar\'e inequality holds for $f\in \Lip(B,  \d)$:
			\begin{equation}
			\int_{B}|f(x)-\E_{\rho_B}(f)|\,\d\rho_B(x)\leq C_4 r_0\int_B |\nabla_x f|\,\d\rho_B,
		\end{equation}
		where $C_4$ depends on $K,N,a_1,a_2$.
	\end{lemma}

	\begin{proof}
		By \cite[Chapter 9]{zbMATH01474795} and \cite[Remark 3.3]{zbMATH06043352},  $\mm$ satisfies the  $(1,1)$-Poincar\'e inequality:
		\begin{equation}
			\int_{B} |f-\E_{\mm_B}(f)|\,\d\mm\leq c_1 r_0 \int_B |\nabla_x f|\,\d\mm,
		\end{equation}
		where $c_1$ depends on $K,N$, $\mm_B=\frac{\mm\restr{B}}{\mm(B)}$. Since $a_1\mm \leq \rho\leq a_2 \mm$, we have
		\begin{equation}
			\begin{aligned}
				\int_{B} |f-\E_{\rho_B}(f)|\,\d\rho\leq &\int_{B} |f-\E_{\mm_B}(f)|\,\d\rho+\int_{B} |\E_{\mm_B}(f)-\E_{\rho_B}(f)|\,\d\rho\\
				\leq & 2a_2\int_{B} |f-\E_{\mm_B}(f)|\,\d\mm\\
				\leq &2a_2 c_1 r_0 \int_B |\nabla_x f|\,\d\mm
				\leq  2\frac{a_2}{a_1}c_1 r_0 \int_B |\nabla_x f|\,\d\rho,\\
			\end{aligned}
		\end{equation}
	which is the thesis.
	\end{proof}

	Next we prove an $L^1$-variant of the gluing Lemma in \cite[Lemma 3.3]{arXiv:2411.04908},   following the same Boman-chain decomposition,  utilizing the doubling property directly instead of the maximal function estimates.

		\begin{definition}[Boman chain condition]\label{Boman chain condition}
		We say that a probability measure $\rho$ on an open set $S$ of a metric space satisfies  \emph{Boman chain condition} with parameters \(E, F,G>1\) if there is a covering \(\mathcal{F}\) of $S$ by open balls  $B \in \mathcal{F}$ such that:

		\begin{enumerate}
				\item For any \(x \in S\),
				\begin{equation*}
						\sum_{B \in \mathcal{F}} \chi_{2B}(x) \leq E \chi_{S}(x).
					\end{equation*}
				\item For some fixed ball \(B_0\) in \(\mathcal{F}\), called the \emph{central ball}, and for every \(B \in \mathcal{F}\), there exists a chain \(B_0, B_1, \ldots, B_N = B\) of distinct balls from \(\mathcal{F}\) such that
				\begin{equation*}
					B \subset F B_j,~~~\forall j \in \{0, \ldots, N-1\}.
					\end{equation*}
			\item Consecutive balls of the above chain overlap quantitatively:
				\begin{equation*}
						\rho(B_j \cap B_{j+1}) \geq G^{-1} \max(\rho(B_j), \rho(B_{j+1})),~~~\forall j \in \{0, \ldots, N-1\}.
					\end{equation*}
			\end{enumerate}
	\end{definition}

	\begin{lemma}\label{3.1}
	For any $f\in L^1(\rho)$,   it holds that
	\begin{equation}\label{A.5}
		\int_S |f(x)-\E_{\rho}(f)|\,\d\rho(x)\leq C_5 \sum_{B\in\mathcal{F}}\rho(B)\int_B |f(x)-\E_{\rho_B}(f)|\,\d\rho_B(x),
	\end{equation}
	where $C_5$ depends on $K,N,\diam(S),S$.
	\end{lemma}

    \begin{proof}
    	Since $S$ is a John domain, and $\rho$ is a doubling measure on $S$, by  \cite[Proposition 3.7]{arXiv:2504.05412}, $\rho$ satisfies the Boman chain condition. Hence there exists a covering $\mathcal{F}$ of $S$ satisfying Definition \ref{Boman chain condition}.

    	For the central ball $B_0$, note that
    	\begin{equation}\label{A.2}
    		\begin{aligned}
    			\int_S |f(x)-\E_{\rho}(f)|\,\d\rho(x)
    			\leq & \int_S |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)+\int_S |\E_{\rho_{B_0}}(f)-\E_{\rho}(f)|\,\d\rho(x)\\
    			\leq & 2\int_S |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x).
    		\end{aligned}
    	\end{equation}

   For  $B\in\mathcal{F}$,  	denote $a_B=\int_B |f-\E_{\rho_{B}}(f)|\,\d\rho=\rho(B)\int_B |f-\E_{\rho_{B}}(f)|\,\d\rho_B$. Since $\mathcal{F}$ is a covering of $S$, we have
    	\begin{equation}\label{A.3}
    		\begin{aligned}
    			&\int_S |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)
    			\leq  \sum_{B \in \mathcal{F}} \int_B |f(x)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)\\
    			\leq & \sum_{B \in \mathcal{F}} \left(\int_B |f(x)-\E_{\rho_{B}}(f)|\,\d\rho(x)+\int_B |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\,\d\rho(x)\right)\\
    			\leq &  \sum_{B \in \mathcal{F}} \left(a_B+\rho(B) |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\right).
    		\end{aligned}
    	\end{equation}

    	For any $B\in\mathcal{F}$, by Boman chain condition, there exists a chain $B_0, B_1,\dots, B_N=B$ of distinct balls from \(\mathcal{F}\), such that for any \(j \in \{0, \ldots, N-1\}\),
    	\begin{equation}
    		\begin{aligned}
    		|\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)|= &\left|\frac{1}{\rho(B_j\cap B_{j+1})}\int_{B_j\cap B_{j+1}}\left(\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)\right)\,\d\rho\right|\\
    		\leq &\frac{1}{\rho(B_j\cap B_{j+1})}\int_{B_j\cap B_{j+1}}\left|\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)\right|\,\d\rho\\
    		\leq &\frac{1}{\rho(B_j\cap B_{j+1})}\left(\int_{B_j\cap B_{j+1}}\left|f-\E_{\rho_{B_j}}(f)\right|+\left|f-\E_{\rho_{B_{j+1}}}(f)\right|\,\d\rho\right)\\
    		\leq & \frac{a_{B_j}+a_{B_{j+1}}}{\rho(B_j\cap B_{j+1})}
    		\overset{*}{\leq}  G \left(\frac{a_{B_j}}{\rho(B_j)}+\frac{a_{B_{j+1}}}{\rho(B_{j+1})}\right),
    		\end{aligned}
    	\end{equation}
    	where $(*)$ follows from the quantitative chain overlap of Boman chain condition $3$.

    	 Thus, we have
    	 \begin{equation}
    	 |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\leq \sum_{j=0}^{N-1}|\E_{\rho_{B_j}}(f)-\E_{\rho_{B_{j+1}}}(f)|\leq 2G \sum_{j=0}^{N}\frac{a_{B_j}}{\rho(B_j)}\overset{*}{\leq} 2G \sum_{B\subset F \bar{B}}\frac{a_{\bar{B}}}{\rho(\bar{B})},
    	 \end{equation}
    	 where $ \sum_{B\subset F \bar{B}}$ means that the sum runs over all  $\bar{B}\in\mathcal{F}$ satisfying $B\subset F \bar{B}$, and $(*)$ follows from the Boman chain condition $2$.
    	 Then by Fubini--Tonelli theorem,
    	 \begin{equation}\label{A.6}
    	 	\begin{aligned}
    	 	 	  \sum_{B \in \mathcal{F}} \rho(B) |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|
    	 	 	  \leq & 2G\sum_{B \in \mathcal{F}} \rho(B) \sum_{B\subset F \overline{B}}\frac{a_{\overline{B}}}{\rho(\overline{B})}
    	 	 	  \leq 2G\sum_{\bar{B} \in \mathcal{F}} \frac{a_{\bar{B}}}{\rho(\bar{B})}\sum_{B\subset F \bar{B}}\rho(B).
    	 	\end{aligned}
    	 \end{equation}

By Boman chain condition $1$ and  the doubling property of $\rho$, we have
    	 \begin{equation}\label{aa}
       \sum_{B\subset F \bar{B}}\rho(B)\leq E\rho(F\bar{B})\leq E\beta^2F^\frac{\log \beta}{\log 2}\rho(\bar{B}),
    	 \end{equation}
where  $\beta=\beta(K,N,\diam(S))$ is the doubling constant.

    	 Combining \eqref{A.2}, \eqref{A.3},  \eqref{A.6} and \eqref{aa}, we obtain
    	 \begin{equation}\label{A_A}
    	 	\begin{aligned}
    	 		 \int_S |f(x)-\E_{\rho}(f)|\,\d\rho(x)\leq & 2 \sum_{B \in \mathcal{F}} \left(a_B+\rho(B) |\E_{\rho_{B}}(f)-\E_{\rho_{B_0}}(f)|\right)\\
    	 		 \leq &2(1+2\beta^2 EF^\frac{\log \beta}{\log 2}G) \sum_{B \in \mathcal{F}} \rho(B)\int_B |f-\E_{\rho_{B}}(f)|\,\d\rho_B,
    	 	\end{aligned}
    	 \end{equation}
  which is the thesis.
    \end{proof}

	\end{appendices}

\begin{theorem}\label{thm0.1}
		Let $(X,\d,\mm)$ be an ${\rm RCD}(K,N)$ metric measure space. Let $S\subseteq X$ be a John domain and $Y\subseteq X$ be compact with $\mm(Y)>0$. Let $\rho\in \mathcal{P}(S)$  be with $a_1\mm\restr{S} \leq \rho\leq a_2\mm\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \diam(S\cup Y)$, such that for any $\mu,\nu\in\mathcal{P}(Y)$,
		\begin{equation}
			\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\leq C W_1^\frac{1}{2}(\mu,\nu),
		\end{equation}
		where $\phi_\mu$ and $\phi_\nu$ are the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
	\end{theorem}

\subsection{Motivation and setting}
	\paragraph{Optimal transport.}Optimal transport, initiated by Monge  \cite{monge1781} and reframed by  Kantorovich \cite{zbMATH03099866},  seeks the most efficient way to redistribute mass between two probability distributions.   Precisely, given two probability measures $\rho$ and $\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a  cost function $c:X\times Y\rightarrow \mathbb{R}$, the Monge optimal transport problem aims to find a minimizer  $T$,   called an optimal transport map,   of the following optimization problem among all measurable maps $H: X \to Y$ pushing $\rho$ forward to $\mu$:
	\begin{equation}\label{2.1}
		\int_X c(x, T(x))  \,{\mathrm d}\rho(x)=\inf_{H_\sharp \rho = \mu}\int_X c(x, H(x))  \,{\mathrm d}\rho(x), \tag{MP}
	\end{equation}
 The   Monge problem  prevents mass splitting, sometimes making the problem ill-posed.  Its relaxation,  called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the  following dual problem:
	\begin{equation}
		\sup_{ \phi(x) + \psi(y) \leq c(x, y)} \left\{ \int_X \phi(x) \, {\mathrm d}\rho(x) + \int_Y \psi(y) \, {\mathrm d}\mu(y)  \right\}.\tag{KD}
	\end{equation}
	The optimal functions $\phi,\psi$,  which always exist under standard assumptions,   are called  Kantorovich potentials \cite{AG-U, V-O}.

	In the Euclidean space,   for $c(x,y)=\frac{1}{2}|x-y|^2$, Brenier's landmark result \cite{brenier1991polar} established that, for absolutely continuous source measures, the optimal transport map is unique and takes the form $T(x)=\nabla u(x)$ for a convex function $u$. This was later extended to Riemannian manifolds by McCann \cite{mccann2001polar}, who showed that for $c(x,y)=\frac{1}{2}{\mathrm d}^2(x,y)$, the optimal transport map is given almost everywhere by $T(x)=\exp_x(-\nabla\varphi(x))$ for a  Kantorovich potential $\varphi$.

	 A fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \cite{LetrouitLecture} and the references therein).   Based on recent breakthroughs in the quantitative stability of optimal transport  on  Euclidean spaces \cite{zbMATH07794624, arXiv:2411.04908},    on boundaries of convex bodies \cite{Kitagawa12,  Kitagawa25}  and  on Riemannian manifolds  \cite{arXiv:2504.05412},
Kitagawa--Letrouit--M\'erigot   \cite[\S 1.2]{arXiv:2504.05412} conjecture that:
	\begin{center}
		\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}
	\end{center}

	\paragraph{General setting.}
	In this paper, we confirm the conjecture of Kitagawa-- Letrouit--M\'erigot for the following spaces:
	\begin{itemize}
		\item  RCD$(K,N)$  spaces (metric measure spaces with synthetic Ricci curvature lower bound):  we prove quantitative $L^1$-stability of Kantorovich potentials with respect to Wasserstein perturbations of the target measure;
		\item Alexandrov spaces with (sectional) curvature lower bound:   we establish stability estimates for optimal transport maps in terms of the $L^2$-norm of the potential difference.
	\end{itemize}

	An Alexandrov space is a  geodesic  space of finite Hausdorff dimension and of curvature bounded from below (cf.  \cite{zbMATH01626771}).   An ${\rm RCD}(K,N)$ space  is  a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition  \cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}.   An $n$-dimensional Alexandrov space with curvature bounded from below by $k$,  equipped with its $n$-Hausdorff measure,   is an ${\rm RCD}(k(n-1), n)$ space  \cite{zbMATH06032507,zbMATH05962558}.
		 After \cite{brenier1991polar, mccann2001polar},   Gigli--Rajala--Sturm \cite[Theorem 1.1]{ gigli2016optimal}  proved the existence and uniqueness of the optimal transport map on an $\rm{RCD}(K,N)$ space $(X,{\mathrm d},\mathfrak m)$,  for the quadratic cost $c(x,y)=\frac{1}{2}{\mathrm d}^2(x,y)$ and the source measure $\rho\ll \mathfrak m$.

	  John domains,    named after  F. John \cite{zbMATH03166766},   encompass many cases of interest, such as  bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \cite{zbMATH00858510} for more discussions).
	\begin{definition}[John domain]
		A bounded open subset $S$ of a metric space is called a \emph{John domain} if there is a distinguished point $x_0 \in S$ and a constant $\eta > 0$ such that, for every $x \in S$, there is a rectifiable curve $\gamma \colon [0, \ell(\gamma)] \to S$ parametrized by arc length, such that $\gamma(0) = x$, $\gamma(\ell(\gamma)) = x_0$, and
		\begin{equation*}
			{\mathrm d}(\gamma(t), S^c) \geq \eta t,~~~\forall t \in [0, \ell(\gamma)],
		\end{equation*}
		where $S^c$ denotes the complement of $S$.
	\end{definition}
	In  \cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\rho$ is the uniform density on some non-John domain  $S\subset \mathbb{R}^n$, then no quantitative stability estimates of the form $\|\phi_\mu-\phi_\nu\|_{L^2(\rho)}\leq C W_p^\alpha(\mu,\nu)$
	can hold for any $C, \alpha, p>0$.
	Moreover, examples in \cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of    $S$ may cause  instability  of optimal transport maps.

	 In this paper,  we always assume that the Kantorovich potential  $\phi_\mu$ from $\rho$ to $\mu$ satisfies $\E_\rho(\phi_\mu)=\int_S\phi_\mu\,{\mathrm d}\rho=0$.  Together with the uniqueness of the optimal transport map, such  $\phi_\mu$ is  unique and thus makes sense to study its stability.

	\subsection{Main results}
	Our main theorem concerns the quantitative  $L^1$ stability of Kantorovich potentials on ${\rm RCD}(K,N)$  spaces.

\subsection{Motivation and setting}
 \paragraph{Optimal transport.}Optimal transport, initiated by Monge \cite{monge1781} and reframed by Kantorovich \cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\rho$ and $\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\times Y\rightarrow \mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \to Y$ pushing $\rho$ forward to $\mu$:
 \begin{equation}\label{2.1}
 \int_X c(x, T(x)) \,{\mathrm d}\rho(x)=\inf_{H_\sharp \rho = \mu}\int_X c(x, H(x)) \,{\mathrm d}\rho(x), \tag{MP}
 \end{equation}
 The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:
 \begin{equation}
 \sup_{ \phi(x) + \psi(y) \leq c(x, y)} \left\{ \int_X \phi(x) \, {\mathrm d}\rho(x) + \int_Y \psi(y) \, {\mathrm d}\mu(y) \right\}.\tag{KD}
 \end{equation}
 The optimal functions $\phi,\psi$, which always exist under standard assumptions, are called Kantorovich potentials \cite{AG-U, V-O}.

A fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \cite{arXiv:2504.05412},
Kitagawa--Letrouit--M\'erigot \cite[\S 1.2]{arXiv:2504.05412} conjecture that:
 \begin{center}
 \emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}
 \end{center}

An Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \cite{zbMATH01626771}). An ${\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\rm RCD}(k(n-1), n)$ space \cite{zbMATH06032507,zbMATH05962558}.
 After \cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\rm{RCD}(K,N)$ space $(X,{\mathrm d},\mathfrak m)$, for the quadratic cost $c(x,y)=\frac{1}{2}{\mathrm d}^2(x,y)$ and the source measure $\rho\ll \mathfrak m$.

John domains, named after F. John \cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \cite{zbMATH00858510} for more discussions).
 \begin{definition}[John domain]
 A bounded open subset $S$ of a metric space is called a \emph{John domain} if there is a distinguished point $x_0 \in S$ and a constant $\eta > 0$ such that, for every $x \in S$, there is a rectifiable curve $\gamma \colon [0, \ell(\gamma)] \to S$ parametrized by arc length, such that $\gamma(0) = x$, $\gamma(\ell(\gamma)) = x_0$, and
 \begin{equation*}
 {\mathrm d}(\gamma(t), S^c) \geq \eta t,~~~\forall t \in [0, \ell(\gamma)],
 \end{equation*}
 where $S^c$ denotes the complement of $S$.
 \end{definition}
 In \cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\rho$ is the uniform density on some non-John domain $S\subset \mathbb{R}^n$, then no quantitative stability estimates of the form $\|\phi_\mu-\phi_\nu\|_{L^2(\rho)}\leq C W_p^\alpha(\mu,\nu)$
 can hold for any $C, \alpha, p>0$.
 Moreover, examples in \cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.

In this paper, we always assume that the Kantorovich potential $\phi_\mu$ from $\rho$ to $\mu$ satisfies $\E_\rho(\phi_\mu)=\int_S\phi_\mu\,{\mathrm d}\rho=0$. Together with the uniqueness of the optimal transport map, such $\phi_\mu$ is unique and thus makes sense to study its stability.

\subsection{Main results}
 Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\rm RCD}(K,N)$ spaces.

John domains, named after F. John \cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \cite{zbMATH00858510} for more discussions).
 \begin{definition}[John domain]
 A bounded open subset $S$ of a metric space is called a \emph{John domain} if there is a distinguished point $x_0 \in S$ and a constant $\eta > 0$ such that, for every $x \in S$, there is a rectifiable curve $\gamma \colon [0, \ell(\gamma)] \to S$ parametrized by arc length, such that $\gamma(0) = x$, $\gamma(\ell(\gamma)) = x_0$, and
 \begin{equation*}
 \d(\gamma(t), S^c) \geq \eta t,~~~\forall t \in [0, \ell(\gamma)],
 \end{equation*}
 where $S^c$ denotes the complement of $S$.
 \end{definition}
 In \cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\rho$ is the uniform density on some non-John domain $S\subset \R^n$, then no quantitative stability estimates of the form $\|\phi_\mu-\phi_\nu\|_{L^2(\rho)}\leq C W_p^\alpha(\mu,\nu)$
 can hold for any $C, \alpha, p>0$.
 Moreover, examples in \cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.

\subsection{Main results}
 Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\rm RCD}(K,N)$ spaces.

If $(S, \d, \mm)$ is a compact $\rcdkn$ space, it supports a global $(1,1)$-Poincar\'e inequality (cf.\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:
 \begin{corollary}\label{cor}
 Let $(X,\d,\mm)$ be a compact ${\rm RCD}(K,N)$ metric measure space. Let $\rho\in \mathcal{P}(X)$ be with $a_1\mm \leq \rho\leq a_2\mm$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \ref{thm0.1} holds.
 \end{corollary}

Adapting the strategy of \cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.
 \begin{theorem}\label{thm0.2}
 Let $(X, \d)$ be an $n$-dimensional Alexandrov space, $\mm$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\rho$ as in Theorem \ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \diam(S\cup Y), {\rm{Per}}(S), S$, such that for any $\mu,\nu\in\mathcal{P}_2(Y)$, we have
 \begin{equation}
 \int_S \d^2(T_\mu(x), T_\nu(x))\,\d \rho(x) \leq CW_1^{1/6}(\mu, \nu),
 \end{equation}
 where $T_\mu$ and $T_\nu$ are the optimal transport maps from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
 \end{theorem}

\begin{proposition}\label{2.11}
 It holds that
 \begin{equation}
 \int_{S}\big| \E_{\mu_x^t[\psi_*]}(v)-\E_{\mu^t[\psi_*]}(v))\big|\,\d\rho(x)\leq \frac{\kappa}{\sqrt{t}}\left(\int_S {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2},
 \end{equation}
 where $\mu^t[\psi_*]$ is defined as in \eqref{mu} and $\kappa$ depends on $K,N,\Lambda_\psi,a_1,a_2, S$.
 \end{proposition}
 \begin{proof}
 Since $a_1\mm\leq \rho\leq a_2\mm$, by \cite[Corollary 2.4]{zbMATH05049052}, $\rho$ is a doubling measure. Moreover,
 since $S$ is a John domain, by \cite[Proposition 3.7]{arXiv:2504.05412}, $\rho$ satisfies the Boman chain condition (see Definition \ref{Boman chain condition}) and we can choose a covering $\mathcal{F}$, such that for any $B\in \mathcal{F}$, $r_B\leq 1$. Then
 \begin{equation}
 \begin{aligned}
 &\int_{S}\big| \E_{\mu_x^t[\psi_*]}(v)-\E_{\mu^t[\psi_*]}(v))\big|\,\d\rho(x)\\
 \mathop{\leq}^{\text {Lemma \ref{3.1}}} &C_5 \sum_{B\in\mathcal{F}}\rho(B)\int_B \big| \E_{\mu_x^t[\psi_*]}(v)-{\rho(B)}^{-1}\E_{\mu^t[\psi_*]}(v))\big|\,\d\rho_B(x)\\
 \mathop{\leq }^{\text{Lemma \ref{p3.10}}}&C_5 \sum_{B\in\mathcal{F}}\rho(B)C_4 r_B\int_B \big|\nabla_x \E_{\mu_x^t[\psi_*]}(v)\big|\,\d\rho_B(x)\\
 \mathop{\leq}^{\text{Proposition \ref{3.9}} }& \frac{C_3C_4C_5}{\sqrt{t}}\sum_{B\in\mathcal{F}}\sqrt{\mm(B_{2r_B})}\sqrt{r_B}\left(\int_B {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2}\\
 \mathop{\leq}^{\text{H\"{o}lder}~~}&\frac{C_3C_4C_5}{\sqrt{t}}\left(\sum_{B\in\mathcal{F}}\mm(B_{2r_B})\right)^\frac{1}{2}\left(\sum_{B\in\mathcal{F}}\int_B {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2}\\
 \overset{\text{Boman chain~}}{\leq}&\frac{\kappa}{\sqrt{t}}\left(\int_S {\rm{Var}}_{ \mu_x^t[\psi_*]}(v)\,\d\rho(x)\right)^\frac{1}{2}
 \end{aligned}
 \end{equation}
 which is the thesis.
 \end{proof}

\begin{theorem}\label{thm2}
 Let $(X, \d)$ be an Alexandrov space with no boundary. Then under the same assumptions of Theorem \ref{thm0.2}, there exists a constant $\bar{C}>0$, depending on $k,n,a_1, a_2, \diam(S\cup Y), {\rm{Per}}(S), S$, such that for any $\mu,\nu\in\mathcal{P}_2(Y)$, we have
 \begin{equation}
 \int_S |\nabla\phi_\mu(x)-\nabla\phi_\nu(x)|^2\,\d\rho(x)\leq \bar{C} \left(\int_S|\phi_\mu(x)-\phi_\nu(x)|^2\,\d\rho(x)\right)^\frac{1}{3},
 \end{equation}
 where $\phi_\mu$ and $\phi_\nu$ are the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
 \end{theorem}

\begin{lemma}[\cite{zbMATH07794624}, Lemma 5.1]\label{l0.18}
 Let $I\subseteq \R$ be a compact segment and let $u,v: I\rightarrow\R$ be two convex functions such that $|u'|$ and $|v'|$ (defined a.e. on $I$) are uniformly bounded on $I$. Then
 \begin{equation*}
 \|u'-v'\|_{L^2(I)}^2\leq 8 (\|u'\|_{L^\infty(I)}+\|v'\|_{L^\infty(I)})^\frac{4}{3}\|u-v\|_{L^2(I)}^\frac{2}{3}.
 \end{equation*}
 \end{lemma}
 Therefore, applying Lemma \ref{l0.18} to the functions $\zeta|s|^2-u_\mu^{(x,v)}$ and $\zeta|s|^2-u_\nu^{(x,v)}$ on each compact segment $[\alpha_i(x,v),\beta_i(x,v)]$ and using H\"{o}lder inequality, we obtain
 \begin{equation}\label{0.55}
 \begin{aligned}
 &\int_S |\nabla\phi_\mu(x)-\nabla\phi_\nu(x)|^2\,\d\rho(x)\\
 \mathop{\leq}^{\eqref{0.54}}& \frac{a_2}{c_nd_n}\int_{{\rm S}X}\sum_{i\in I_S(x,v)}\int_{\alpha_i(x,v)}^{\beta_i(x,v)}\langle\nabla\phi_\mu(b_s(x,v))-\nabla\phi_\nu(b_s(x,v)),t_s(x,v)\rangle^2\,\d s \,\d\mm_S(x,v)\\
 \leq&C_1 \int_{{\rm S}X}\sum_{i\in I_S(x,v)}\left(\int_{\alpha_i(x,v)}^{\beta_i(x,v)}|\phi_\mu(b_s(x,v))-\phi_\nu(b_s(x,v))|^2\,\d s\right)^\frac{1}{3}\,\d\mm_S(x,v)\\
 \leq&C_1 \int_{{\rm S}X}\left(\# I_S\right)^\frac{2}{3}\left(\int_0^1|\phi_\mu(b_s)-\phi_\nu(b_s)|^2\chi_S(b_s)\,\d s\right)^\frac{1}{3}\d\mm_S\\
 \leq&C_1 \left(\int_{{\rm S}X}\# I_S\,\d\mm_S\right)^\frac{2}{3}\left(\int_{{\rm S}X}\int_0^1|\phi_\mu(b_s)-\phi_\nu(b_s)|^2\chi_S(b_s)\,\d s \,\d\mm_S\right)^\frac{1}{3}\\
 \leq&C_1 \left(\frac{c_n}{a_1}\right)^\frac{1}{3}\left(\int_{{\rm S}X}\# I_S\,\d\mm_S\right)^\frac{2}{3}\left(\int_S|\phi_\mu-\phi_\nu|^2\,\d\rho\right)^\frac{1}{3},
 \end{aligned}
 \end{equation}
 where $C_1$ depends on $k,n,a_2, \diam(S\cup Y)$, and the last inequality follows from the invariance of $\mm_S$ under the geodesic flow.

Let $\phi_\mu$ and $\phi_\nu$ be the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively. From \cite{zbMATH05349267,zbMATH06032507} we know that $\nabla \phi_\mu(x), \nabla \phi_\nu(x) \in {\rm{T}}_xX$ and
 $T_\mu(x)= \exp_x(-\nabla \phi_\mu(x))$, $T_\nu(x)= \exp_x(-\nabla \phi_\nu(x))$ are well-defined for almost every $x\in S$. By triangle comparison condition (cf. \cite{zbMATH05342782}), it holds
 \begin{equation}\label{exp}
 \d \big(T_\mu(x), T_\nu(x) \big) \leq C_2|\nabla \phi_\mu(x)-\nabla \phi_\nu(x)|,
 \end{equation}
 for some constant $C_2>0$ which depends on $k, \diam(S\cup Y)$.
 \medskip

If $(S,  {\mathrm d},  \mathfrak m)$  is a compact ${\rm RCD}(K, N)$ space,   it  supports a global $(1,1)$-Poincar\'e inequality (cf.\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:
	\begin{corollary}\label{cor}
		Let $(X,{\mathrm d},\mathfrak m)$ be a compact ${\rm RCD}(K,N)$ metric measure space.  Let $\rho\in \mathcal{P}(X)$ be with $a_1\mathfrak m \leq \rho\leq a_2\mathfrak m$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \ref{thm0.1} holds.
	\end{corollary}

	Adapting the  strategy of \cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.
		\begin{theorem}\label{thm0.2}
		Let $(X,  {\mathrm d})$ be an $n$-dimensional Alexandrov space, $\mathfrak m$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\rho$ as in Theorem \ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \mathop{\rm diam}\nolimits(S\cup Y), {\rm{Per}}(S), S$, such that for any $\mu,\nu\in\mathcal{P}_2(Y)$, we have
		\begin{equation}
			\int_S {\mathrm d}^2(T_\mu(x), T_\nu(x))\,{\mathrm d} \rho(x) \leq CW_1^{1/6}(\mu, \nu),
		\end{equation}
		where $T_\mu$ and $T_\nu$ are the optimal transport maps from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.
	\end{theorem}

	\subsection{Strategy: heat kernel-regularized \texorpdfstring{$c$}{c}-transform}

Motivated by   regularized $c$-transforms  using  Gibbs kernels $e^{-c(x, y)/\varepsilon}$ \cite{arXiv:2504.05412},  entropic optimal transport \cite{GTJEMS, GTT25}  and Varadhan's formula
\[
\lim_{t \to 0} -t \log p_{t/2}(x, y) = \frac{1}{2} {\mathrm d}^2(x, y)=c(x,y),
\]
  we make use of {heat kernel-regularized $c$-transform} (Hopf--Cole transform):
\[
\mathop{\rm Lip}\nolimits(X, {\mathrm d}) \ni \psi  \mapsto \Phi_t[\psi](x) = -t \log \int_X e^{\frac{\psi(y)}{t}} p_{t/2}(x, y) \, {\mathrm d}\mathfrak m(y).
\]
This approach  allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.

We define the heat kernel regularized Kantorovich functional as
	\begin{equation*}
		\K_t[\varphi]:=\int_S \Phi_t[\varphi]\,{\mathrm d}\rho.
	\end{equation*}
Similar to \cite{arXiv:2504.05412},  the key in our proof is the strong concavity of the functional $\K_t$.
To achieve this,  we  first derive a local strong concavity estimate using heat kernel estimate,    then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.

Unlike the regularized $c$-transform used in \cite{arXiv:2504.05412},  the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument.  This possibility is ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.

\medskip

	\noindent {\bf Organization.}  This paper is structured as follows. In Section~\ref{sec3}, we prove quantitative stability of Kantorovich potentials on $\mathrm{RCD}(K,N)$ spaces. In Section \ref{sec4}, we establish stability of optimal transport maps on Alexandrov spaces. The Appendix \ref{A} contains technical lemmas about Poincar\'e inequalities.

	\bigskip

\noindent {\bf Declaration.} The authors declare no conflict of interest and that the manuscript has no associated data.

\medskip

\noindent {\bf Acknowledgement.} This work is supported by the Ministry of Science \& Technology of China (2021YFA1000900, 2021YFA1002200),  National Natural Science Foundation of China  (12201596)   and Shandong Provincial Natural Science Foundation (ZR2025QB05).  The authors  thank Nicola Gigli, Jun Kitagawa, Nan Li,  Cyril Letrouit, Quentin M\'erigot, Luca Tamanini for  helpful discussions and  suggestions on the bibliography.

Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, the proof of Theorem~\ref{thm0.1} uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)
\[
\psi\mapsto \Phi_t[\psi](x)=-t\log\int_X e^{\frac{\psi(y)}{t}}\,p_{t/2}(x,y)\,\mathrm d\mathfrak m(y),
\]
which “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”
\[
\mathcal K_t[\varphi]:=\int_S \Phi_t[\varphi]\,\mathrm d\rho.
\]
“Similar to \cite{arXiv:2504.05412}, the key in our proof is the strong concavity of the functional $\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”

They also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”

Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, in establishing the main theorem, the proof uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)
\[
\psi\mapsto \Phi_t[\psi](x)=-t\log\int_X e^{\frac{\psi(y)}{t}}\,p_{t/2}(x,y)\,\mathrm d\mathfrak m(y),
\]
which “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”
\[
\mathcal K_t[\varphi]:=\int_S \Phi_t[\varphi]\,\mathrm d\rho.
\]
“Similar to arXiv:2504.05412, the key in our proof is the strong concavity of the functional $\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”

They also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”,

\label{thm0.1}
		Let $(X,{\mathrm d},\mathfrak m)$ be an ${\rm RCD}(K,N)$ metric measure space. Let $S\subseteq X$ be a John domain and $Y\subseteq X$ be compact with $\mathfrak m(Y)>0$. Let $\rho\in \mathcal{P}(S)$  be with $a_1\mathfrak m\restr{S} \leq \rho\leq a_2\mathfrak m\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \mathop{\rm diam}\nolimits(S\cup Y)$, such that for any $\mu,\nu\in\mathcal{P}(Y)$,
		\begin{equation}
			\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}\leq C W_1^\frac{1}{2}(\mu,\nu),
		\end{equation}
		where $\phi_\mu$ and $\phi_\nu$ are the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $\nu$ respectively.,

Question:
Let \((X,{\mathrm d},\mathfrak m)\) be an \({\rm RCD}(K,N)\) metric measure space. Let \(S\subseteq X\) be a John domain, meaning that \(S\) is a bounded open set for which there exist \(x_0\in S\) and \(\eta>0\) such that every \(x\in S\) can be joined to \(x_0\) by a rectifiable curve \(\gamma:[0,\ell(\gamma)]\to S\), parametrized by arc length, with \(\gamma(0)=x\), \(\gamma(\ell(\gamma))=x_0\), and \({\mathrm d}(\gamma(t),X\setminus S)\ge \eta t\) for all \(t\in[0,\ell(\gamma)]\). Let \(Y\subseteq X\) be compact with \(\mathfrak m(Y)>0\). Assume \(\rho\in \mathcal P(S)\) satisfies \(a_1\,\mathfrak m\!\restriction_S \le \rho \le a_2\,\mathfrak m\!\restriction_S\) for some constants \(a_1,a_2>0\). For each \(\mu\in\mathcal P(Y)\), let \(\phi_\mu\) denote a Kantorovich potential for the quadratic cost \(c(x,y)=\tfrac12{\mathrm d}(x,y)^2\) from \(\rho\) to \(\mu\), normalized by \(\int_S \phi_\mu\,{\mathrm d}\rho=0\). Also let
\[
W_1(\mu,\nu):=\inf_{\pi\in\Pi(\mu,\nu)}\int_{Y\times Y}{\mathrm d}(y,z)\,{\mathrm d}\pi(y,z)
\]
be the 1-Wasserstein distance. Which quantitative estimate holds for all \(\mu,\nu\in\mathcal P(Y)\)?

There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu)^{1/2}.
\]

Choices:
B.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu).
\]
C.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C.
\]
D.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2\), such that for every John domain \(S\subseteq X\), every compact \(Y\subseteq X\) with \(\mathfrak m(Y)>0\), and every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu)^{1/2}.
\]
E.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^\infty(S)}\le C\,W_1(\mu,\nu)^{1/2}.
\]

The stem states hypotheses and asks for the resulting uniform stability estimate, but it does not explicitly reveal the correct exponent, norm, or quantifier structure. There is no direct answer leakage beyond the standard setup.

The item is close to a theorem-recall question: it essentially asks for the exact conclusion under a detailed set of assumptions. The answer is not literally restated in the stem, but the task is mainly to recognize the theorem's precise formulation.

Some reasoning is required to distinguish the sharp statement from nearby variants involving a stronger Lipschitz rate, weaker boundedness-only claim, dependence of constants on the measures, or an incorrect L^infty norm. However, this is mostly discriminating among theorem variants rather than generating a conclusion from first principles.

The distractors are mathematically plausible and target common errors: replacing the sharp 1/2 exponent by 1, dropping the Wasserstein dependence, weakening uniformity via target-dependent constants, and strengthening L^1 to L^infty. They are distinct and well aligned with likely failure modes.

A solid theorem-discrimination MCQ with strong distractors and no obvious leakage, but it mainly tests precise recall/recognition of a known result rather than deeper generative reasoning.


Existential–Universal
Inequality or Bound

Question:
Let \((X,{\mathrm d},\mathfrak m)\) be an \({\rm RCD}(K,N)\) metric measure space. Let \(S\subseteq X\) be a John domain, meaning that \(S\) is a bounded open set for which there exist \(x_0\in S\) and \(\eta>0\) such that every \(x\in S\) can be joined to \(x_0\) by a rectifiable curve \(\gamma:[0,\ell(\gamma)]\to S\), parametrized by arc length, with \(\gamma(0)=x\), \(\gamma(\ell(\gamma))=x_0\), and \({\mathrm d}(\gamma(t),X\setminus S)\ge \eta t\) for all \(t\in[0,\ell(\gamma)]\). Let \(Y\subseteq X\) be compact with \(\mathfrak m(Y)>0\). Assume \(\rho\in \mathcal P(S)\) satisfies \(a_1\,\mathfrak m\!\restriction_S \le \rho \le a_2\,\mathfrak m\!\restriction_S\) for some constants \(a_1,a_2>0\). For each \(\mu\in\mathcal P(Y)\), let \(\phi_\mu\) denote a Kantorovich potential for the quadratic cost \(c(x,y)=\tfrac12{\mathrm d}(x,y)^2\) from \(\rho\) to \(\mu\), normalized by \(\int_S \phi_\mu\,{\mathrm d}\rho=0\). Also let
\[
W_1(\mu,\nu):=\inf_{\pi\in\Pi(\mu,\nu)}\int_{Y\times Y}{\mathrm d}(y,z)\,{\mathrm d}\pi(y,z)
\]
be the 1-Wasserstein distance. Which quantitative estimate holds for all \(\mu,\nu\in\mathcal P(Y)\)?

Choices:
B.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu).
\]
C.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C.
\]
D.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2\), such that for every John domain \(S\subseteq X\), every compact \(Y\subseteq X\) with \(\mathfrak m(Y)>0\), and every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^1(\rho)}=\int_S |\phi_\mu-\phi_\nu|\,{\mathrm d}\rho \le C\,W_1(\mu,\nu)^{1/2}.
\]
E.
There exists a constant \(C>0\), depending only on \(K,N,a_1,a_2,S\), and \(\operatorname{diam}(S\cup Y)\), such that for every \(\mu,\nu\in\mathcal P(Y)\),
\[
\|\phi_\mu-\phi_\nu\|_{L^\infty(S)}\le C\,W_1(\mu,\nu)^{1/2}.
\]

The stem does not explicitly state the target estimate or reveal the correct option. It gives the hypotheses and asks for the valid conclusion, without obvious verbal cues favoring A.

This is essentially a theorem-recall item: the stem lays out the full hypotheses and asks which estimate holds. The correct answer is basically the theorem's conclusion rather than a genuinely new inference from competing principles.

Some discrimination is required among exponent, norm, and parameter-dependence choices, especially between A and the plausible stronger/weaker variants. However, the task is primarily recognition/recall of the sharp statement, not substantial generative reasoning.

The distractors are mathematically meaningful and target common failure modes: overstrong Lipschitz rate (B), weaker but non-sharp bound (C), incorrect uniformity in constants (D), and an unjustified upgrade from L1 to L∞ control (E).

A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it is mostly a direct restatement of a known result rather than a deep reasoning question.


The systematic study of multiple zeta values (MZVs) started
in the early 1990s with the work of Hoffman \cite{[H-PJM]}
and Zagier \cite{[Z-FECM]}. Let $r\ge 1$ be an integer. For any multiple index
$\mathbf{s}=(s_1,s_2,\cdots,s_r)\in \mathbb{Z}_{\geq 1}^r$ with
$s_r\textgreater 1$, the {\it multiple zeta value (MZV for short)}
is defined by
\begin{align*}
\zeta(s_1,s_2,\cdots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r}
\frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}}.
\end{align*}
The {\it weight} of MZV is defined to be the number
$|\mathbf{s}|:=s_1+\cdots+s_r$, and its {\it depth} is defined
by $l(\mathbf{s}):=r$. For brevity, we let $\{a_1,\cdots,a_k\}^d$
denote the sequence formed by repeating the sequence
$\{a_1,\cdots,a_k\}$ exactly $d$ times. The simplest precise
evaluations of MZVs and MZSVs are given by
\begin{eqnarray*}
\zeta(\{2\}^d)=\frac{\pi^{2d}}{(2d+1)!}.
\end{eqnarray*}

A central problem in MZV is to investigate the algebraic structure
of the $\mathbb{Q}$-vector space $\mathfrak{Z}_k$ spanned by MZVs
of weight $k$. In this direction, Hoffman \cite{[H-JA]} proposed a
conjecture that stats that every MZV of weight $k$ can be expressed as a
$\mathbb{Q}$-linear combination of those MZVs whose entries are only
$2$'s and $3$'s. This conjecture was proved by Brown \cite{[B-ANN]}
using motivic method. In particular, Brown showed that
$H(a,b):=\zeta(\{2\}^a,3,\{2\}^b)$ can be expressed as a $\mathbb{Q}$-linear
combination of the products $\pi^{2m}\zeta(2n+1)$ with $m+n=a+b+1$.
In 2012, Zagier \cite{[Z-AM]} found surprisingly an elegant and explicit
formula as follows.

\section{Introduction}
The systematic study of multiple zeta values (MZVs) started
in the early 1990s with the work of Hoffman \cite{[H-PJM]}
and Zagier \cite{[Z-FECM]}. Let $r\ge 1$ be an integer. For any multiple index
$\mathbf{s}=(s_1,s_2,\cdots,s_r)\in \mathbb{Z}_{\geq 1}^r$ with
$s_r\textgreater 1$, the {\it multiple zeta value (MZV for short)}
is defined by
\begin{align*}
\zeta(s_1,s_2,\cdots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r}
\frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}}.
\end{align*}
The {\it weight} of MZV is defined to be the number
$|\mathbf{s}|:=s_1+\cdots+s_r$, and its {\it depth} is defined
by $l(\mathbf{s}):=r$. For brevity, we let $\{a_1,\cdots,a_k\}^d$
denote the sequence formed by repeating the sequence
$\{a_1,\cdots,a_k\}$ exactly $d$ times. The simplest precise
evaluations of MZVs and MZSVs are given by
\begin{eqnarray*}
\zeta(\{2\}^d)=\frac{\pi^{2d}}{(2d+1)!}.
\end{eqnarray*}

A central problem in MZV is to investigate the algebraic structure
of the $\mathbb{Q}$-vector space $\mathfrak{Z}_k$ spanned by MZVs
of weight $k$. In this direction, Hoffman \cite{[H-JA]} proposed a
conjecture that stats that every MZV of weight $k$ can be expressed as a
$\mathbb{Q}$-linear combination of those MZVs whose entries are only
$2$'s and $3$'s. This conjecture was proved by Brown \cite{[B-ANN]}
using motivic method. In particular, Brown showed that
$H(a,b):=\zeta(\{2\}^a,3,\{2\}^b)$ can be expressed as a $\mathbb{Q}$-linear
combination of the products $\pi^{2m}\zeta(2n+1)$ with $m+n=a+b+1$.
In 2012, Zagier \cite{[Z-AM]} found surprisingly an elegant and explicit
formula as follows.

In a similar way, Hoffman \cite{[H-CNTP]} defined the multiple
$t$-values (``odd variant'' of MZVs). For any multiple index
$\mathbf{s}=(s_1,s_2,...,s_r)\in \mathbb{Z}_{\geq 1}^r$, the
{\it multiple $t$-value } is defined by
\begin{align*}
t(s_1,s_2,...,s_r):=\sum_{1\leq k_1<k_2<...<k_r}
\frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}},
\end{align*}
where if the index is empty, then we define the value $t(\emptyset):=1$.
As we stated above, first let us recall the the analogue formula
for the simplest multiple zeta value (\cite{[MK-PSPM]},\cite{[Z-FM]})
\begin{align*}
t(\mathop{\underbrace{2,2,...,2}}_{n})=\frac{\pi^{2n}}{2^{2n}(2n)!}.
\end{align*}

By the method of establishing the generating functions, Murakami \cite{[M-MA]}
obtained an equivalent version of Theorem \ref{theorem1.01}. Define
$$\widetilde{t}(k_1,k_2,...,k_r):=2^{|k|}t(k_1,k_2,...,k_r)$$
and set
$$K(a,b):=\widetilde{t}(\{2\}^a,3,\{2\}^b),K(n)=\widetilde{t}(\{2\}^n)$$
for any $a,b,n\in \mathbb{N}_{\geq 0}$. Then for all integers $a,b\geq 0$,
Murakami \cite{[M-MA]} provided the following formula
\begin{align}\label{eq1.1}
K(a,b)=\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\zeta(2k+1),
\end{align}
where
$$d_{a,b}:=\binom{2k}{2a+1}+\binom{2k}{2b+1}\Big(1-\frac{1}{2^{2k}}\Big).$$
Moreover, let $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. Then \eqref{eq1.1} is equivalent to
\begin{align}
T(a,b)=2\sum_{k=1}^{r+s+1}(-1)^{k-1}d_{a,b}\frac{1}{2^{2k}}\zeta(2k+1)
t(\{2\}^{a+b-k+1}),
\end{align}

\begin{conjecture}\label{con2}
For nonnegative integers $a$ and $b$, we have
\begin{align}
T(a,b)=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align}
\end{conjecture}
\noindent Notice that Conjectures 1.2 and 1.3 are shown in \cite{[L-MZ]}
to be true when $b=0$. But for the general case $b>0$, these two conjectures
are still kept open so far.

\begin{theorem}\label{theorem1.1}
For nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where
\begin{align*}
L(a,b):=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty}
\frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}.
\end{align*}
\end{theorem}

\begin{theorem}\label{theorem1.2}
For nonnegative integers $a$ and $b$, we have $T(a,b)=\hat L(a,b)$, where
\begin{align*}
\hat L(a,b):=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align*}
\end{theorem}

First, we consider the integral
$$\int_{0}^{\frac{\pi}{2}} x^{2a + 2}\left(1 - \frac{2}{\pi} x\right)^{2b + 1} \cot(x)\mathrm{d}x.$$
By using the well-known identity (see, for example, Theorem 1.2.4 of \cite{[AAR]})
\begin{align}\label{eq3.1'}
x\cot(x)=-2\sum_{n=0}^{\infty}\frac{\zeta(2n)}{\pi^{2n}}x^{2n},
\end{align}
we derive that
\begin{align}\label{eq3.2}
&\int_{0}^{\frac{\pi}{2}}x^{2a+2}\Big(1-\frac{2}{\pi} x\Big)^{2b+1}
\cot(x)\mathrm{d}x\nonumber\\
=&-2\int_{0}^{\frac{\pi}{2}}x^{2a+1}\Big(1-\frac{2}{\pi} x\Big)^{2b+1}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{\pi^{2n}}x^{2n} \mathrm{d}x \nonumber\\
=&-2\sum_{n=0}^{\infty}\frac{\zeta(2n)}{\pi^{2n}}
\int_{0}^{\frac{\pi}{2}}x^{2n+2a+1}\Big(1-\frac{2}{\pi} x\Big)^{2b+1}\mathrm{d}x.
\end{align}

\begin{theorem}\label{theorem1.01}
(Zagier \cite{[Z-AM]}) For non-negative integers $a$ and $b$,
\begin{align}
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\zeta(2k+1)
\zeta(\{2\}^{a+b+1-k}),
\end{align}
where
$$c_{a,b}:=\binom{2k}{2a+2}-\Big(1-\frac{1}{2^{2k}}\Big)\binom{2k}{2b+1}.$$
\end{theorem}

\noindent The proof of Theorem \ref{theorem1.01} is given by computing the associated
generating functions of both sides in a closed form, and then showing
they are entire functions of exponential growth that agree at
sufficiently many points to force their equality. Later on, his proof
was simplified by Li \cite{[L-MRL]}. In 2017, Hessami Pilehrood
and Hessami Pilehrood \cite{[HHT-JMAA]} gave an alternative proof of Theorem
\ref{theorem1.01}.

In a similar way, Hoffman \cite{[H-CNTP]} defined the multiple
$t$-values  (``odd variant'' of MZVs). For any multiple index
$\mathbf{s}=(s_1,s_2,...,s_r)\in \mathbb{Z}_{\geq 1}^r$, the
{\it multiple $t$-value } is defined by
\begin{align*}
t(s_1,s_2,...,s_r):=\sum_{1\leq k_1<k_2<...<k_r}
\frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}},
\end{align*}
where if the index is empty, then we define the value $t(\emptyset):=1$.
As we stated above, first let us recall the the analogue formula
for the simplest multiple zeta value (\cite{[MK-PSPM]},\cite{[Z-FM]})
\begin{align*}
t(\mathop{\underbrace{2,2,...,2}}_{n})=\frac{\pi^{2n}}{2^{2n}(2n)!}.
\end{align*}

By the method of establishing the generating functions, Murakami \cite{[M-MA]}
obtained an equivalent version of Theorem \ref{theorem1.01}. Define
$$\widetilde{t}(k_1,k_2,...,k_r):=2^{|k|}t(k_1,k_2,...,k_r)$$
and set
$$K(a,b):=\widetilde{t}(\{2\}^a,3,\{2\}^b),K(n)=\widetilde{t}(\{2\}^n)$$
for any $a,b,n\in \mathbb{N}_{\geq 0}$. Then for all integers $a,b\geq 0$,
Murakami \cite{[M-MA]} provided the following formula
\begin{align}\label{eq1.1}
K(a,b)=\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\zeta(2k+1),
\end{align}
where
$$d_{a,b}:=\binom{2k}{2a+1}+\binom{2k}{2b+1}\Big(1-\frac{1}{2^{2k}}\Big).$$
Moreover, let $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. Then \eqref{eq1.1} is equivalent to
\begin{align}
T(a,b)=2\sum_{k=1}^{r+s+1}(-1)^{k-1}d_{a,b}\frac{1}{2^{2k}}\zeta(2k+1)
t(\{2\}^{a+b-k+1}),
\end{align}

In a different approach, Lupu \cite{[L-MZ]} provided elementary proofs of
Zagier's formula for the special case $b=0$, and also established a Zagier-type
formula for multiple $t$-values when $b=0$. For the general case $b>0$, Lupu
\cite{[L-MZ]} proposed the following two conjectures.

\begin{conjecture}\label{con1}
For nonnegative integers $a$ and $b$, we have
\begin{align}
H(a,b)=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty}
\frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}.
\end{align}
\end{conjecture}

\begin{conjecture}\label{con2}
For nonnegative integers $a$ and $b$, we have
\begin{align}
T(a,b)=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align}
\end{conjecture}
\noindent Notice that Conjectures 1.2 and 1.3 are shown in \cite{[L-MZ]}
to be true when $b=0$. But for the general case $b>0$, these two conjectures
are still kept open so far.

In 2026, Lai, Lupu, and Orr \cite{[LLO-PAMS]} gave elementary and direct proofs
for both $H(a, b)$ and $T(a, b)$. In fact, they \cite{[LLO-PAMS]} first provided
an integral expression for $H(a, b)$, and then showed that this integral expression
is equal to Zagier's result. The same proof method was applied to $T(a, b)$.

In this paper, our main goal is to investigate Conjectures \ref{con1} and \ref{con2}.
We will prove that both conjectures of Lupu are true. Actually, by employing the integral
expressions presented in \cite{[LLO-PAMS]} for $H(a,b)$ and $T(a,b)$ and using
the properties of Beta and Gamma functions, we show the following main results
of this paper.

\begin{theorem}\label{theorem1.1}
For nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where
\begin{align*}
L(a,b):=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty}
\frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}.
\end{align*}
\end{theorem}

\begin{theorem}\label{theorem1.2}
For nonnegative integers $a$ and $b$, we have $T(a,b)=\hat L(a,b)$, where
\begin{align*}
\hat L(a,b):=\frac{-2}{(2a+1)!}\left(\frac{\pi}{2}\right)^{2a+2b+2}
\sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}.
\end{align*}
\end{theorem}

This paper is organized as follows. In Section 2, we supply several preliminary lemmas
that are needed in the proofs of our main results. Sections 3 and 4 are devoted to
the proofs of Theorems \ref{theorem1.1} and \ref{theorem1.2}, respectively.

The post-theorem introduction sketches Zagier's strategy for proving Theorem~\ref{theorem1.01}: compute the associated generating functions of both sides "in a closed form," then show these generating functions are "entire functions of exponential growth" and that they "agree at sufficiently many points to force their equality." It is also noted that "his proof was simplified by Li" and that Hessami Pilehrood and Hessami Pilehrood gave "an alternative proof" of Theorem~\ref{theorem1.01}.

The post-theorem introduction sketches Zagier's strategy for proving the main theorem: compute the associated generating functions of both sides "in a closed form," then show these generating functions are "entire functions of exponential growth" and that they "agree at sufficiently many points to force their equality." It is also noted that "his proof was simplified by Li" and that Hessami Pilehrood and Hessami Pilehrood gave "an alternative proof" of the main theorem.

Question:
Let \(a,b\in \mathbb Z_{\ge 0}\). For a multiple index \((s_1,\dots,s_r)\) with \(s_r>1\), define the multiple zeta value
\[
\zeta(s_1,\dots,s_r)=\sum_{1\le k_1<\cdots<k_r}\frac{1}{k_1^{s_1}\cdots k_r^{s_r}}.
\]
Let \(\{2\}^m\) denote the index consisting of \(m\) copies of \(2\), and define
\[
H(a,b):=\zeta(\{2\}^a,3,\{2\}^b).
\]
Which statement holds for every such pair \((a,b)\)?

For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^k\left[\binom{2k}{2a+2}-\left(1-\frac{1}{2^{2k}}\right)\binom{2k}{2b+1}\right]\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}).
\]
Equivalently, in the \(k\)-th summand the coefficient is \(c_{a,b}(k)=\binom{2k}{2a+2}-\left(1-2^{-2k}\right)\binom{2k}{2b+1}\).

Choices:
B.
For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=0}^{a+b+1}(-1)^k\left[\binom{2k}{2a+2}-\left(1-\frac{1}{2^{2k}}\right)\binom{2k}{2b+1}\right]\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}).
\]
Equivalently, the same coefficient \(c_{a,b}(k)=\binom{2k}{2a+2}-\left(1-2^{-2k}\right)\binom{2k}{2b+1}\) is used, but the sum starts at \(k=0\).
C.
For all nonnegative integers \(a\) and \(b\), the value \(H(a,b)\) is a \(\mathbb{Q}\)-linear combination of products of the form
\[
\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}) \qquad (1\le k\le a+b+1).
\]
In particular, \(H(a,b)\) can be written as a \(\mathbb{Q}\)-linear combination of \(\pi^{2m}\zeta(2n+1)\) with \(m+n=a+b+1\).
D.
For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^k\left[\binom{2k}{2a+2}-\left(1-\frac{1}{2^{2k}}\right)\binom{2k}{2b+1}\right]\zeta(2k+1)\,\zeta(\{2\}^{a+b-k}).
\]
Equivalently, the coefficient is \(c_{a,b}(k)=\binom{2k}{2a+2}-\left(1-2^{-2k}\right)\binom{2k}{2b+1}\), but the depth-\((a+b-k)\) block of twos appears in the second zeta factor.
E.
For all nonnegative integers \(a\) and \(b\),
\[
H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^k\left[\binom{2k}{2a+2}-\binom{2k}{2b+1}\right]\zeta(2k+1)\,\zeta(\{2\}^{a+b+1-k}).
\]
Equivalently, the coefficient in the \(k\)-th summand is \(c_{a,b}(k)=\binom{2k}{2a+2}-\binom{2k}{2b+1}\).

explicit coefficient formula and exact closed-form identity

weight-balancing index in the second zeta factor

essential factor \((1-2^{-2k})\) in the coefficient

The stem only defines the notation and asks which statement is universally true; it does not reveal the specific identity, coefficients, or range of summation.

The item is largely a recognition task for the precise theorem statement about H(a,b), though it does include competing nearby alternatives rather than simply restating the theorem in the stem.

There is some reasoning pressure: one can reject distractors by checking weight balance, the lower summation bound, and the missing factor \((1-2^{-2k})\). However, success still depends mostly on recalling the exact identity rather than generating a conclusion from premises.

The distractors are mathematically plausible and target realistic errors: wrong summation boundary, weaker true statement, weight/index mismatch, and omission of an essential coefficient factor.

A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recognition rather than deeper generative mathematical reasoning.


\label{thm 1}
Let $\Omega\subset \mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\in C^2([0, +\infty))$ satisfies  $h'(r)\ge 0$ and $h''(r)\ge 0$ for $r\in (0, +\infty)$. Let  $B \subset \mathbb{R}^m$  be the  origin-centered  round ball with the same $\gamma_h$-volume as $\Omega$, namely $ \int_{\Omega} e^{h(|x|)}\, dx= \int_{B} e^{h(|x|)}\, dx$. Denote by $\sigma_1(B; \gamma_h)$ the first nonzero Steklov eigenvalue of \eqref{eq 1.4} on $B$, and $\l_{2,\alpha}(\Omega; \gamma_h)$ the second Robin eigenvalue of \eqref{eq 1.1} on $\Omega$.  If  $\alpha \in [-\sigma_1(B; \gamma_h), 0]$, then
\begin{align}\label{eq 1.6}
\l_{2,\alpha}(\Omega; \gamma_h) \le \l_{2,\alpha}(B; \gamma_h).
\end{align}
Equality holds  if and only if
$\Omega=B$.

\begin{theorem}\label{thm 1}
Let $\Omega\subset \R^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\in C^2([0, +\infty))$ satisfies  $h'(r)\ge 0$ and $h''(r)\ge 0$ for $r\in (0, +\infty)$. Let  $B \subset \R^m$  be the  origin-centered  round ball with the same $\gamma_h$-volume as $\Omega$, namely $ \int_{\Omega} e^{h(|x|)}\, dx= \int_{B} e^{h(|x|)}\, dx$. Denote by $\sigma_1(B; \gamma_h)$ the first nonzero Steklov eigenvalue of \eqref{eq 1.4} on $B$, and $\l_{2,\a}(\Omega; \gamma_h)$ the second Robin eigenvalue of \eqref{eq 1.1} on $\Omega$.  If  $\a \in [-\sigma_1(B; \gamma_h), 0]$, then
\begin{align}\label{eq 1.6}
\l_{2,\a}(\Omega; \gamma_h) \le \l_{2,\a}(B; \gamma_h).
\end{align}
Equality holds  if and only if
$\Omega=B$.
\end{theorem}

The classical Szeg\"o-Weinberger inequality \cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian among all bounded domains of the same volume in Euclidean space. Using the arguments of Szeg\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings: Ashbaugh-Benguria \cite{AB95} extended the the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space.
Moreover, it is expected (cf. \cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\"o \cite{FL20} and the Weinberger \cite{FL21} approaches.
For further results on Szeg\"o-Weinberger type inequalities, we refer to \cite{AS96, As99, BBC20, BCB16, LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.

Eigenvalue optimization problems for the weighted Laplacian have also attracted considerable attention. For instance, Chiacchio and di Blasio \cite{CB12} proved that, among all origin-symmetric regions with fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \cite{BCB16} obtained optimal Szeg\"o-Weinberger type inequalities under a weighted measure.
Additional results on weighted eigenvalue problems can be found in \cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.

Inspired by the work of Brock, Francesco and di Blasio \cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen \cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems
\begin{align}\label{eq 1.1}
 \begin{cases}
 -\operatorname{div} (e^{h(|x|)}\nabla u)=\lambda e^{h(|x|)} u, \quad & x\in \Omega,\\
 \frac{\partial u}{\partial \nu}+\alpha u=0, \quad & x\in \partial \Omega,
 \end{cases}
\end{align}
where $\alpha\in \mathbb{R}$. Here and in the sequel, $\Omega$ denotes a bounded connected domain in $\mathbb{R}^m$ with Lipchitz boundary, $\nu$ is the outward normal to $\partial \Omega$, and $h(r)$ is a $C^2$ function on $[0, +\infty)$. We define the weighted measure associated with $h$ by
$$
d\gamma_h=e^{h(|x|)}\,dx, \qquad x\in \mathbb{R}^m,
$$
and denote by $W^{1,2}(\Omega; \gamma_h)$ the weighted Sobolev space equipped with the norm
\begin{align*}
||u||_{W^{1,2}(\Omega;\gamma_h)}:=\left(\int_\Omega u^2 \, d\gamma_h\right)^{1/2}+ \left(\int_\Omega |\nabla u|^2 \, d\gamma_h\right)^{1/2}.
\end{align*}

It is well known that the problem \eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\l_{k, \alpha}(\Omega; \gamma_h)$ for $k=1,2, \cdots$, which satisfy
 \begin{align*}
 \l_{1, \alpha}(\Omega;\gamma_h) < \l_{2, \alpha}(\Omega;\gamma_h) \leq \l_{3, \alpha}(\Omega;\gamma_h) \leq \cdots \to +\infty,
 \end{align*}
with each eigenvalue repeated according to its multiplicity.
By standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \eqref{eq 1.1} admit the variational characterizations:
\begin{align}
 \lambda_{1,\alpha}(\Omega; \gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}\, dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}\},
\end{align}
where $dA$ is the
induced measure on $\partial \Omega$;
and
\begin{align}\label{eq 1.3}
 \lambda_{2,\alpha}(\Omega;\gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}, \int_{\Omega}uu_{1}d\gamma_h=0\},
\end{align}
where $u_1(x)$ is an eigenfunction corresponding to $\l_{1, \alpha}(\Omega;\gamma_h)$.

To state our main results, we introduce the following Steklov eigenvalue problem on $\Omega$
\begin{align}\label{eq 1.4}
\begin{cases}
 -\operatorname{div} (e^{h(|x|)}\nabla u)=0 \quad & x\in \Omega,\\
 \frac{\partial u}{\partial \nu}=\sigma u, \quad & x\in \partial \Omega.
\end{cases}
\end{align}
The first nonzero Steklov eigenvalue can be characterized variationally by
\begin{align}
\sigma_1(\Omega; \gamma_h)=\inf\left\{ \frac{\int_{\Omega} |\nabla u|^2 \, d\gamma_h}{\int_{\partial \Omega} u^2 e^{h}\, dA} : u\in W^{1,2}(\Omega; \gamma_h )\setminus\{0\},\;\; \int_{\partial \Omega} ue^{h(|x|)}\ dA=0 \right\}.
\end{align}
 In this paper, we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \eqref{eq 1.1}.

\begin{align}\label{eq 1.1}
    \begin{cases}
        -\div (e^{h(|x|)}\nabla u)=\l e^{h(|x|)} u, \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}+\a u=0, \quad & x\in \p \Omega,
    \end{cases}
\end{align}

\begin{align}\label{eq 1.4}
\begin{cases}
      -\div (e^{h(|x|)}\nabla u)=0 \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}=\sigma u, \quad & x\in \p \Omega.
\end{cases}
\end{align}

It is well known that the problem \eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\l_{k, \a}(\Omega; \gamma_h)$ for $k=1,2, \cdots$, which satisfy
 \begin{align*}
 \l_{1, \a}(\Omega;\gamma_h) < \l_{2, \a}(\Omega;\gamma_h) \leq \l_{3, \a}(\Omega;\gamma_h) \leq \cdots \to +\infty,
 \end{align*}
with each eigenvalue repeated according to its multiplicity.
By standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \eqref{eq 1.1} admit the variational characterizations:
\begin{align}
 \lambda_{1,\alpha}(\Omega; \gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}\, dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}\},
\end{align}
where $dA$ is the
induced measure on $\p \Omega$;
and
\begin{align}\label{eq 1.3}
 \lambda_{2,\alpha}(\Omega;\gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}, \int_{\Omega}uu_{1}d\gamma_h=0\},
\end{align}
where $u_1(x)$ is an eigenfunction corresponding to $\l_{1, \a}(\Omega;\gamma_h)$.

By taking $\a=-\sigma_1(B;\gamma_h)$, Theorem \ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \cite[Corollary 1.2]{MZ24}.
\begin{corollary}\label{cor}
Under the hypotheses of Theorem \ref{thm 1}, we have
\begin{align*}
\sigma_1(\Omega; \gamma_h)\le \sigma_1(B; \gamma_h).
\end{align*}
Equality holds if and only if $\Omega$ is isometric to the round ball $B$.
\end{corollary}

\begin{proposition} \label{prop 2.4}
Suppose $h'(r)\ge 0$, and $\a\ge -\sigma_1(B(R); \gamma_h)$. Then
\begin{align*}
 \lambda_{2,\alpha}(B(R); \gamma_h)\geq 0.
\end{align*}
\end{proposition}
\begin{proof}
 Let $g(r)$ be as defined in Proposition \ref{prop 2.3}.
 Since
 \begin{align*}
 \int_{\partial B(R)}g(r)\frac{x_{i}}{r}e^{h(r)}dA=0, 1\leq i\leq m,
 \end{align*}
 the functions $u_{i}=g(r)\frac{x_{i}}{r}, 1\leq i\leq m,$ are admissible test functions for $\sigma_{1}(B(R), \gamma_h)$. Therefore,
 \begin{align}\label{eq 2.22}
 \sum_{i=1}^{m}\int_{B(R)}|\nabla u_{i}|^{2}\, d\gamma_h\geq \sigma_1(B(R); \gamma_h) \sum_{i=1}^{m}\int_{\partial B(R)}|u_{i}|^{2}e^{h(r)}dA,
 \end{align}
 where $dA$ is the induced measure on $\p B$.
Using
 \begin{align*}
 \sum_{i=1}^{m}|\nabla u_{i}|^{2}=g'(r)^{2}+\frac{m-1}{r^{2}}g(r)^{2}.
 \end{align*}
 and the fact $h(r)\le h(R)$ for $r\in[0, R]$, we estimate from \ref{eq 2.22}
 \begin{align}\label{eq 2.23}
 0&\leq \sum_{i=1}^{m}\int_{B(R)}|\nabla u_{i}|^{2}e^{h(r)}dx- \sigma_1(B(R); \gamma_h)\sum_{i=1}^{m}\int_{\partial B(R)}|u_{i}|^{2}e^{h(r)}dA\nonumber\\
 &\leq \omega_{m-1}\int_{0}^{R}\left(g'(r)^{2}+\frac{m-1}{r^{2}}g^{2}\right)e^{h(r)}r^{m-1}dr+\alpha \omega_{m-1} g(R)^{2}e^{h(R)}R^{m-1},
 \end{align}
 where $\omega_{m-1}$ denotes the surface area of the unit sphere $\mathbb{S}^{m-1}$ in $\R^m$.
Using integration by parts and \eqref{eq 2.11}, we compute
\begin{align*}
 &\int_{0}^{R}\left(g'(r)^{2}+\frac{m-1}{r^{2}}g^{2}\right)e^{h(r)}r^{m-1}dr\\
 =\quad &gg'e^{h(r)}r^{m-1}\Big|_{0}^{R}-\int_{0}^{R}g(g'e^{h(r)}r^{m-1})'dr+\int_{0}^{R}\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\
 =\quad &g(R)g'(R)e^{h(R)}R^{m-1}-\int_{0}^{R}g(g''+(\frac{m-1}{r}+h')g')e^{h(r)}r^{m-1}dr\\
 &+\int_{0}^{R}\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\
 =&g(R)g'(R)e^{h(R)}R^{m-1}+\int_{0}^{R}g^{2}(\lambda_{2,\alpha}(B(R);\gamma_h)-\frac{m-1}{r^{2}})e^{h(r)}r^{m-1}dr\\
 &+\int_{0}^{R}\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr.
\end{align*}
Substituting this equality into inequality \eqref{eq 2.23} yields
 \begin{align*}
 0\leq& \lambda_{2,\alpha}(B(R);\gamma_h)\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}dr+g(R)e^{h(R)}R^{m-1}(g'(R)+\alpha g(R))\\
 =&\lambda_{2,\alpha}(B(R);\gamma_h)\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}\, dr,
 \end{align*}
which implies $\lambda_{2,\alpha}(B(R);\gamma_h)\geq 0$.
\end{proof}

From now on, we assume $\Omega\subset \R^m$ is a bounded Lipschitz domain symmetric about the origin, and $B\subset \R^m$ is the origin-centered round bound with the same $\gamma_h$ voulme as $\Omega$. Let $R$ be the radius of $B$. We extend the function $g$ defined in \eqref{eq 2.11} by
\begin{equation}\label{eq 3.1}
g(r)=g(R) e^{-\a(r-R)}\text{\quad for \quad} r\ge R.
\end{equation}
By definition, $g$ is continuously differentiable on $(0,\infty)$. If $\a\le 0$, then by Proposition \ref{prop 2.2}, $g$ is increasing on $(0,\infty)$. In the sequel, $\sigma_1(B; \gamma_h)$ denotes the first nonzero Steklov eigenvalue of $B$ as defined in \eqref{eq 2.2}.

{\bf Case 2.} $r\ge R$. For $r\ge R$,
\begin{align*}
 F(r)=(-\alpha^{2}+\frac{m-1}{r^{2}}+\alpha\frac{m-1}{r}+\alpha h^{'})g^{2}(r),
\end{align*}
and a direct computation yields
\begin{align*}
 F'(r)=(2\alpha^{3}+\alpha h''-2\alpha^{2}h'-2\alpha^{2}\frac{m-1}{r}-3\alpha \frac{m-1}{r^{2}}-2\frac{m-1}{r^{3}})g^{2}(r).
\end{align*}
Since $\alpha\leq 0$, $h'\geq 0$ and $h^{''}\geq 0$, we obtain
\begin{align*}
 F'(r)\le& \left(-2\alpha^{2}\frac{m-1}{r}-3\alpha \frac{m-1}{r^{2}}-2\frac{m-1}{r^{3}}\right) g(r)^2\\
 =&-\frac{m-1}{r}(2(\alpha+\frac{3}{4r})^{2}+\frac{7}{8}\frac{1}{r^{2}})g(r)^2<0,
\end{align*}
proving that $F(r)$ is monotonically decreasing on $(0, R)$.
\end{proof}
 The main idea in proving Theorem \ref{thm 1} is to construct trial functions for $\l_{2, \a}$ (see \eqref{eq 1.3}) using techniques introduced by Weinberger. We first recall a monotonicity lemma for weighted symmetrization.
\begin{lemma}\label{Lemma 3.2}
 Let $\Omega\subset \R^m$ be a bounded set, and let $B$ be the origin-centered round ball with the same $\gamma_h$ volume as $\Omega$. If $f(r)$ is monotonically decreasing on $[0,+\infty)$, then
 \begin{align*}
 \int_\Omega f(|x|)\, d\gamma_m\le \int_B f(|x|)\, d\gamma_m.
 \end{align*}
 Equality holds if and only if $\Omega=B$.
\end{lemma}
\begin{proof}
 Lemma 3.2 follows directly from Hardy-Littlewood inequality; see the proof of inequality (4.24) in \cite[Page 213]{BCB16}.
\end{proof}

Now we now proceed to prove Theorem \ref{thm 1}.
\begin{proof}
 Let $\Omega$ be a bounded Lipschitz domain symmetric about the origin in $\R^m$, and let $B\subset \R^m$ be the origin-centered round ball of radius $R$ satisfying
 $$
\int_B \, d\gamma_h =\int_\Omega \, d\gamma_h.
 $$
Define $g(r)$ for $r\in [0, +\infty)$ as in \eqref{eq 3.1}, and for each $1\le i\le m$ set
\begin{align*}
 v_i(x)=g(|x|)\frac{x_i}{|x|},
\end{align*}
where $x=(x_1, x_2, \cdots, x_m)\in \R^m$.

By taking $\alpha=0$, Theorem \ref{thm 1} implies that the ball centered at the
origin is the unique set maximizer the second Neumann eigenvalue $\mu_2(\Omega;\gamma_h)$ among all bounded Lipschitz domains $\Omega$ in $\mathbb{R}^m$ with prescribed $\gamma_h$-measure and symmetric about the origin. This result was previously  proved by Brock, Chiacchio, and di Blasio (Theorem 1.2 of \cite{BCB16}). Note that Theorem \ref{thm 1} applies in particular  to $\l_{2, \alpha}(\Omega; e^{|x|^2/2})$. Hence, our result provides information about  eigenvalues of the problem
$$
-\Delta u-x\cdot \nabla u=\lambda u
$$
which has been widely studied in literature (see, e.g. \cite{BCM12, BMP13}).
The symmetry assumption on $\Omega$ in Theorem \ref{thm 1}
 ensures the validity of  the orthogonality conditions \eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is  radially
 symmetric about the origin.
It remains an interesting question that whether \eqref{eq 1.6}
holds in the case of the Gauss measure $e^{-|x|^2/2}dx$.

By taking $\alpha=-\sigma_1(B;\gamma_h)$, Theorem \ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \cite[Corollary 1.2]{MZ24}.
\begin{corollary}\label{cor}
Under the hypotheses of Theorem \ref{thm 1}, we have
\begin{align*}
\sigma_1(\Omega; \gamma_h)\le \sigma_1(B; \gamma_h).
\end{align*}
Equality holds if and only if $\Omega$ is isometric to the round ball $B$.
\end{corollary}

The main technique in the proof of Theorem \ref{thm 1} relies on Weinberger’s trick, in which  test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties in the proof include analyzing the properties of the second eigenfunction of \eqref{eq 1.1} on the round ball  and establishing the required monotonicity to apply Weinberger’s method. The rest of the paper is organized as follows.
In Section \ref{sect2}, we study properties of the Robin and Steklov eigenvalue  problems for balls in the weighted space. In Section  \ref{sect3}, we prove Theorem \ref{thm 1} and Corollary \ref{cor}.

The proof of Theorem~\ref{thm 1} "relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues." The "main difficulties" are "analyzing the properties of the second eigenfunction of \eqref{eq 1.1} on the round ball" and "establishing the required monotonicity to apply Weinberger’s method." The symmetry assumption on $\Omega$ "ensures the validity of the orthogonality conditions \eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin."

The proof of the main theorem relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties are analyzing the properties of the second eigenfunction of
\begin{align}\label{eq 1.1}
    \begin{cases}
        -\div (e^{h(|x|)}\nabla u)=\l e^{h(|x|)} u, \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}+\a u=0, \quad & x\in \p \Omega,
    \end{cases}
\end{align}
on the round ball and establishing the required monotonicity to apply Weinberger’s method. The symmetry assumption on $\Omega$ ensures the validity of the orthogonality conditions
\begin{align}\label{eq 3.5}
        \int_\Omega v_i(x)u(x)   \, d\gamma_h=0, \quad i=1,2,\cdots, m.
    \end{align}
as the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin.

\label{thm 1}
Let $\Omega\subset \mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\in C^2([0, +\infty))$ satisfies  $h'(r)\ge 0$ and $h''(r)\ge 0$ for $r\in (0, +\infty)$. Let  $B \subset \mathbb{R}^m$  be the  origin-centered  round ball with the same $\gamma_h$-volume as $\Omega$, namely $ \int_{\Omega} e^{h(|x|)}\, dx= \int_{B} e^{h(|x|)}\, dx$. Denote by $\sigma_1(B; \gamma_h)$ the first nonzero Steklov eigenvalue of
\begin{align}\label{eq 1.4}
\begin{cases}
      -\div (e^{h(|x|)}\nabla u)=0 \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}=\sigma u, \quad & x\in \p \Omega.
\end{cases}
\end{align}
on $B$, and $\l_{2,\alpha}(\Omega; \gamma_h)$ the second Robin eigenvalue of
\begin{align}\label{eq 1.1}
    \begin{cases}
        -\div (e^{h(|x|)}\nabla u)=\l e^{h(|x|)} u, \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}+\a u=0, \quad & x\in \p \Omega,
    \end{cases}
\end{align}
on $\Omega$.  If  $\alpha \in [-\sigma_1(B; \gamma_h), 0]$, then
\begin{align}\label{eq 1.6}
\l_{2,\alpha}(\Omega; \gamma_h) \le \l_{2,\alpha}(B; \gamma_h).
\end{align}
Equality holds  if and only if
$\Omega=B$.,

Question: Let Ω ⊂ R^m be a bounded Lipschitz domain symmetric about the origin, and let h ∈ C^2([0,∞)) satisfy h'(r) ≥ 0 and h''(r) ≥ 0 for all r > 0. Define the weighted measure dγ_h = e^{h(|x|)} dx. Let B ⊂ R^m be the origin-centered round ball with the same γ_h-volume as Ω, i.e. ∫_Ω e^{h(|x|)} dx = ∫_B e^{h(|x|)} dx. For a domain D, let λ_{2,α}(D; γ_h) denote the second eigenvalue of the weighted Robin problem -div(e^{h(|x|)}∇u) = λ e^{h(|x|)}u in D, with boundary condition ∂u/∂ν + αu = 0 on ∂D. Let σ_1(B; γ_h) denote the first nonzero Steklov eigenvalue on B for -div(e^{h(|x|)}∇u) = 0 in B, with ∂u/∂ν = σu on ∂B. If α ∈ [-σ_1(B; γ_h), 0], which quantitative estimate holds?

One has λ_{2,α}(Ω; γ_h) ≤ λ_{2,α}(B; γ_h), and equality holds if and only if Ω = B.

Choices:
B. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\) for every \(\alpha \le 0\), and equality holds if and only if \(\Omega = B\).
C. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\).
D. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\) whenever \(\alpha \in [-\sigma_1(\Omega; \gamma_h),0]\), and equality holds if and only if \(\Omega = B\).
E. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \ge \lambda_{2,\alpha}(B; \gamma_h)\), and equality holds if and only if \(\Omega = B\).

sharp range for \alpha tied to monotonicity/Steklov threshold

dependence of the threshold on B rather than on \Omega

extremal direction from Weinberger trial-function argument

The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the valid conclusion, without giving away the inequality direction or the precise equality condition.

This is essentially a theorem-recall item: the stem lists the exact hypotheses and asks for the corresponding conclusion. The correct choice is basically the theorem statement itself.

Some reasoning or careful discrimination is needed because the options vary in subtle but important ways (range of α, equality characterization, comparison direction). However, the task is closer to recognition of the precise theorem than to genuine derivation.

The distractors are strong and plausible: one overextends the α-range, one gives only a weaker true conclusion, one swaps the Steklov threshold to Ω, and one reverses the inequality. These reflect realistic mathematical confusions.

A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests precise theorem recall/recognition rather than generative reasoning.


Question: Let Ω ⊂ R^m be a bounded Lipschitz domain symmetric about the origin, and let h ∈ C^2([0,∞)) satisfy h'(r) ≥ 0 and h''(r) ≥ 0 for all r > 0. Define the weighted measure dγ_h = e^{h(|x|)} dx. Let B ⊂ R^m be the origin-centered round ball with the same γ_h-volume as Ω, i.e. ∫_Ω e^{h(|x|)} dx = ∫_B e^{h(|x|)} dx. For a domain D, let λ_{2,α}(D; γ_h) denote the second eigenvalue of the weighted Robin problem -div(e^{h(|x|)}∇u) = λ e^{h(|x|)}u in D, with boundary condition ∂u/∂ν + αu = 0 on ∂D. Let σ_1(B; γ_h) denote the first nonzero Steklov eigenvalue on B for -div(e^{h(|x|)}∇u) = 0 in B, with ∂u/∂ν = σu on ∂B. If α ∈ [-σ_1(B; γ_h), 0], which quantitative estimate holds?

Choices:
B. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\) for every \(\alpha \le 0\), and equality holds if and only if \(\Omega = B\).
C. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\).
D. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\) whenever \(\alpha \in [-\sigma_1(\Omega; \gamma_h),0]\), and equality holds if and only if \(\Omega = B\).
E. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \ge \lambda_{2,\alpha}(B; \gamma_h)\), and equality holds if and only if \(\Omega = B\).

The stem does not reveal the conclusion directly; it only states the hypotheses and asks which estimate is valid. The correct choice is not leaked by wording, though the setup strongly suggests a known extremal theorem.

This is essentially a theorem-recall item: the stem reproduces the full hypothesis set and asks for the exact conclusion. The correct option is basically the theorem statement itself.

Some reasoning is needed to distinguish the exact sharp statement from nearby variants: stronger range in B, weaker conclusion in C, wrong threshold domain in D, and reversed inequality in E. But it mainly tests precise recall rather than substantial generative mathematical reasoning.

The distractors are strong and mathematically meaningful: overgeneralized parameter range, omission of the equality characterization, wrong dependence on the Steklov threshold, and reversed extremal direction. These reflect realistic failure modes.

A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it is largely a direct restatement of a known result rather than a deep reasoning question.


\begin{Theorem}\label{ILR2n}
Let $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then, for every embedding $f$ of $K$ into ${\mathbb R}^{2n}$, there exists an element $\lambda$ in $\Lambda^{n-1,n}(K)$ such that ${\rm lk}_{2}(f(\lambda)) = 1$.
\end{Theorem}

Throughout this paper, we work in the piecewise linear category. For the fundamentals of piecewise-linear topology, we refer the reader to \cite{H69}, \cite{RS82}. Let $K$ be a finite simplicial $n$-complex, which we identify with its polyhedron in this context. For an integer $k\le n$, let $\varDelta^{k}(K)$ denote the set of all $k$-simplices in $K$. Then the subcomplex $\bigcup_{l\le k}\varDelta^{l}(K)$ of $K$ is called the {\it $k$-skeleton} of $K$ and denoted by $K^{k}$. Let $f$ be an embedding of $K$ into $\mathbb{R}^{m}$. Let $\Lambda^{p,q}(K)$ be the set of all unordered pairs $\{\gamma, \gamma'\}$ of mutually disjoint subcomplexes of $K$ such that $\gamma$ is homeomorphic to the $p$-sphere and $\gamma'$ is homeomorphic to the $q$-sphere. We identify any pair $(\gamma,\gamma')$ in $\Lambda^{p,q}(K)$ with the disjoint union $\gamma\sqcup \gamma'$. Then, for any $\lambda = \gamma \sqcup \gamma' \in \Lambda^{p,q}(K)$, the image $f(\lambda) = f(\gamma) \sqcup f(\gamma')$ forms a two-component link in $f(K)$ consisting of a $p$-sphere and a $q$-sphere. If $p+q = m-1$, the \textit{$\mathbb{Z}_2$-linking number} ${\rm lk}_2(L) = {\rm lk}_2(S^p, S^q) = {\rm lk}_2(S^q, S^p) \in \mathbb{Z}_2$ of a two-component link $L = S^p \sqcup S^q$ in ${\mathbb R}^m$ is well-defined (cf.\ \cite{ST80}). If $m=2n+1$, there are known examples of simplicial $n$-complexes with the following property: every embedding into $\mathbb{R}^{2n+1}$ necessarily contains a two-component link of $n$-spheres with $\mathbb{Z}_2$-linking number $1$. Such examples were constructed by Segal--Spie\.{z} \cite{SeSp92}, Lovasz--Schrijver \cite{LS98}, Skopenkov \cite{Sk03}, Taniyama \cite{T00}, and more recently by the author \cite{N26}. In particular, the case of $n=1$ is famous as the Conway--Gordon--Sachs theorem (\cite{CG83}, \cite{S84}), and such a simplicial $1$-complex (i.e., a graph) is said to be {\it intrinsically linked}.

Our purpose in this paper is to present an example of a minimal simplicial $n$-complex that possesses such an intrinsic linking property with respect to embeddings into $\mathbb{R}^{2n}$. Let us introduce three simplicial $n$-complexes $N_{1}^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ as follows. First, let $\sigma_{m}=|a_{0}a_{1}\cdots a_{m}|$ be an $m$-simplex whose vertices are $a_{0},a_{1},\ldots,a_{m}$. This can be naturally regarded as a simplicial $m$-complex consisting of itself and all its faces. In this setting, the $k$-skeleton of $\sigma_{m}$ is standardly denoted by $\sigma_{m}^{k}$ for $k\le m$. Then we define
\begin{eqnarray*}
N_{1}^{(n)} = \sigma_{2n+2}^{n}\setminus \{|a_{0}a_{j_{1}}\cdots a_{j_{n}}|\ |\ 1\le j_{1}<j_{2}<\cdots<j_{n}\le 2n+2\}.
\end{eqnarray*}
In other words, $N_{1}^{(n)}$ is the subcomplex of $\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices that contain the vertex $a_0$. Next, for a positive integer $k$, let $[k]^{*m}$ denote the $m$-fold join of $k$ points. Explicitly, we set
\begin{eqnarray*}
[k]^{*m} =
\{a_{0}^{0},a_{1}^{0},\ldots,a_{k-1}^{0}\} * \{a_{0}^{1},a_{1}^{1},\ldots,a_{k-1}^{1}\} * \cdots * \{a_{0}^{n},a_{1}^{n},\ldots,a_{k-1}^{n}\}.
\end{eqnarray*}
Note that $[k]^{*n+1}$ can naturally be regarded as a simplicial $n$-complex. Then we define
\begin{eqnarray*}
N_{2}^{(n)} = [3]^{*n+1}\setminus \{|a_{0}^{0}a_{j_{1}}^{1}\cdots a_{j_{n}}^{n}|\ |\ j_{1},j_{2},\ldots,j_{n}\in\{0,1,2\}\}.
\end{eqnarray*}
In other words, $N_{2}^{(n)}$ is the subcomplex of $[3]^{*n+1}$ obtained by removing all $n$-simplices that contain the vertex $a_{0}^{0}$. Finally, we define
\begin{eqnarray*}
N_{3}^{(n)} = \sigma_{2n+2}^{n}\setminus \varDelta^{n}(\sigma_{n+1}).
\end{eqnarray*}
In other words, $N_{3}^{(n)}$ is the subcomplex of $\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices contained in $\sigma_{n+1}=|a_{0}a_{1}\cdots a_{n+1}|$. As we will see in the next section, it is well known that neither $\sigma_{2n+2}^{n}$ nor $[3]^{*n+1}$ cannot be embedded in ${\mathbb R}^{2n}$ (Theorem \ref{VKF}). On the other hand, it is also known that any proper subcomplex of $\sigma_{2n+2}^{n}$ and of $[3]^{*n+1}$ can be embedded in $\mathbb{R}^{2n}$ (Gr\"{u}nbaum \cite{Grun69}). Therefore, each of $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ is embeddable in $\mathbb{R}^{2n}$. Then we have the following:

\begin{Theorem}{\rm (van Kampen \cite{VK33}, Flores \cite{Flores32})}\label{VKF}
Let $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\varphi$ of $K$ into ${\mathbb R}^{2n}$, the following holds:
\begin{eqnarray*}
\sum_{\substack{s,s'\in\varDelta^{n}(K)\\ s\cap s'=\varnothing}}l(\varphi(s),\varphi(s'))\equiv 1\pmod{2}.
\end{eqnarray*}
\end{Theorem}

Therefore, $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ are simplicial $n$-complexes that are `intrinsically linked' with respect to embeddings into ${\mathbb R}^{2n}$. We note that the simplicial $n$-complex $\sigma_{2n+2}^{n}\setminus\{|a_{0}a_{1}\cdots a_{n}|\}$ has already been shown to satisfy this property (Freedman--Krushkal \cite[Proposition 4.1]{FK14}; see also Freedman--Krushkal--Teichner \cite[Lemma 6]{FKT94}). However, both our $N_{1}^{(n)}$ and $N_{3}^{(n)}$ are proper subcomplexes of this complex. Moreover, the complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ are minimal in the following sense among all simplicial $n$-complexes possessing this property.

\begin{Proposition}\label{min}
Let $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\to {\mathbb R}^{2n}$ such that ${\rm lk}_{2}(f(\lambda))= 0$ for any element $\lambda$ in $\Lambda^{n-1,n}(N)$.
\end{Proposition}

\begin{Remark}\label{nsp}
A graph $G$ is said to be {\it non-separating planar} if there exists an embedding $f: G \to \mathbb{R}^2$ such that, for every cycle $\gamma$ in $G$, all vertices of $G$ not lying on $\gamma$ belong to the same connected component of $\mathbb{R}^2 \setminus f(\gamma)$. Equivalently, $G$ is not non-separating planar if and only if for every embedding $f:G\to \mathbb{R}^2$, the image $f(G)$ contains a two-component link consisting of a $0$-sphere and a $1$-sphere with $\mathbb{Z}_2$-linking number $1$. Dehkordi--Farr proved in \cite{DF} that a graph is non-separating planar if and only if it contains no subgraph homeomorphic to $K_1 \sqcup K_4$, $K_1 \sqcup K_{2,3}$, or $K_{1,1,3}$ as illustrated in Fig. \ref{nonsp}. They further applied this characterization to the study of linkless embeddings of graphs in $\mathbb{R}^3$. For related results, see Fowler--Li--Pavelescu \cite{FLP23}, Pavelescu--Pavelescu \cite{PP24}. Note that $N_1^{(1)}$ is isomorphic to $K_1 \sqcup K_4$, $N_2^{(1)}$ is isomorphic to $K_1 \sqcup K_{2,3}$, and $N_3^{(1)}$ is isomorphic to $K_{1,1,3}$. Thus, our simplicial $n$-complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ may be regarded as higher-dimensional analogues of the minimal subgraphs that prevent a graph from being non-separating planar.
\end{Remark}

\begin{Theorem}{\rm (van Kampen \cite{VK33}, Flores \cite{Flores32})}\label{VKF}
Let $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\varphi$ of $K$ into ${\mathbb R}^{2n}$, the following holds:
\begin{eqnarray*}
\sum_{\substack{s,s'\in\varDelta^{n}(K)\\ s\cap s'=\varnothing}}l(\varphi(s),\varphi(s'))\equiv 1\pmod{2}.
\end{eqnarray*}
\end{Theorem}

\begin{proof}[Proof of Theorem \ref{ILR2n}]
We first consider the case $K = N_1^{(n)}$. In what follows, for an $m$-simplex $s$, we will also denote its $(m-1)$-skeleton $s^{m-1}$ by $\partial s$. We define collections $\Gamma^{n-1}$ and $\Gamma^n$ of $(n-1)$ and $n$-subcomplexes of $N_1^{(n)}$, respectively, as follows:
\begin{eqnarray*}
\Gamma^{n-1} &=& \left\{\partial s \mid s\in \varDelta^{n}(\sigma_{2n+2}^{n}),\,a_{0}\in s\right\},\\
\Gamma^{n} &=& \left\{\partial t \mid t\in \varDelta^{n+1}(\sigma_{2n+2}),\,a_{0}\notin t\right\}.
\end{eqnarray*}
Then every $\lambda \in \Lambda^{n-1,n}(N_1^{(n)})$ is the disjoint union of an element $\gamma$ of $\Gamma^{n-1}$ and an element $\gamma'$ of $\Gamma^n$, see Fig. \ref{N_1_link} for $n=2$. Let $f:N_{1}^{(n)}\to {\mathbb R}^{2n}$ be an embedding. This can be extended to a generic immersion $\hat{f}:\sigma_{2n+2}^{n} \to \mathbb{R}^{2n}$ such that, for any two $n$-simplices $s,s' \in \varDelta^n(\sigma_{2n+2}^{n})$ with $a_0 \in s$ and $a_0 \notin s'$, the images $\hat{f}(s)$ and $\hat{f}(s')$ intersect transversely at finitely many double points, all of which occur in the interiors of both $n$-simplices. Note that for any $\lambda = \gamma \sqcup \gamma' \in \Lambda^{n-1,n}(N_1^{(n)})$ with $\gamma = \partial s$, $s \in \varDelta^n(\sigma_{2n+2}^n)$ and $a_0 \in s$, we have
\begin{eqnarray}\label{lkdef}
{\rm lk}_{2}(f(\gamma),f(\gamma'))
\equiv \sum_{s'\in \varDelta^{n}(\gamma')}l(\hat{f}(s),\hat{f}(s'))\pmod{2}.
\end{eqnarray}
On the other hand, for any two disjoint $n$-simplices $s, s' \in \varDelta^n(\sigma_{2n+2}^n)$ with $a_0 \in s$ and $a_0 \notin s'$, there uniquely exists an element $\lambda = \gamma \sqcup \gamma' \in \Lambda^{n-1,n}(N_1^{(n)})$ such that $\gamma = \partial s$ and $\gamma' \supset s'$. Then, by (\ref{lkdef}) and Theorem \ref{VKF}, we obtain
\begin{eqnarray*}\label{lksum}
\sum_{\lambda\in \Lambda^{n-1,n}(N_{1}^{(n)})}{\rm lk}_{2}(f(\lambda))
&=& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{1}^{(n)}) \\ \gamma\in \Gamma^{n-1},\,\gamma'\in\Gamma^{n}}}{\rm lk}_{2}(f(\gamma),f(\gamma'))\\
&\equiv& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{1}^{(n)}) \\ \gamma=\partial s,\,s\in \varDelta^{n}(\sigma_{2n+2}^{n}),\,a_{0}\in s}}
\bigg(\sum_{s'\in \varDelta^{n}(\gamma')}l(\hat{f}(s),\hat{f}(s'))\bigg)\nonumber\\
&=& \sum_{\substack{s,\,s'\in \varDelta^{n}(\sigma_{2n+2}^{n}) \\ s\cap s'=\varnothing}}
l(\hat{f}(s),\hat{f}(s'))\nonumber\\
&\equiv& 1\pmod{2}. \nonumber
\end{eqnarray*}
Thus, there exists some $\lambda$ in $\Lambda^{n-1,n}(N_{1}^{(n)})$ such that ${\rm lk}_{2}(f(\lambda))=1$.

Next, we consider the case $K = N_{2}^{(n)}$. We define collections $\Gamma^{n-1}$ and $\Gamma^n$ of $(n-1)$ and $n$-subcomplexes of $N_{2}^{(n)}$, respectively, as follows:
\begin{eqnarray*}
\Gamma^{n-1} &=& \left\{\partial s \mid s\in \varDelta^{n}([3]^{*n+1}),\,a_{0}^{0}\in s\right\},\\
\Gamma^{n} &=& \left\{\left\{a_{1}^{0},a_{2}^{0}\right\}*\left\{a_{i_{1}}^{1},a_{j_{1}}^{1}\right\}*\cdots*\left\{a_{i_{n}}^{n},a_{j_{n}}^{n}\right\} \mid 0\le i_{q}<j_{q}\le 2,\,q\ge 1\right\}.
\end{eqnarray*}
Note that the $(n+1)$-fold join of two points is homeomorphic to the $n$-sphere. Then every $\lambda \in \Lambda^{n-1,n}(N_{2}^{(n)})$ is the disjoint union of an element $\gamma$ of $\Gamma^{n-1}$ and an element $\gamma'$ of $\Gamma^n$, see Fig. \ref{N_2_link} for $n=2$. Let $f:N_{2}^{(n)}\to {\mathbb R}^{2n}$ be an embedding. This can be extended to a generic immersion $\hat{f}:[3]^{*n+1} \to \mathbb{R}^{2n}$ such that, for any two $n$-simplices $s,s' \in \varDelta^n([3]^{*n+1})$ with $a_{0}^{0} \in s$ and $a_{0}^{0} \notin s'$, the images $\hat{f}(s)$ and $\hat{f}(s')$ intersect transversely at finitely many double points, all of which occur in the interiors of both $n$-simplices. Then, in a similar way as in the case of $N_{1}^{(n)}$, we have
\begin{eqnarray*}\label{lksum2}
\sum_{\lambda\in \Lambda^{n-1,n}(N_{2}^{(n)})}{\rm lk}_{2}(f(\lambda))
&=& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{2}^{(n)}) \\ \gamma\in \Gamma^{n-1},\,\gamma'\in\Gamma^{n}}}{\rm lk}_{2}(f(\gamma),f(\gamma'))\\
&\equiv& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{2}^{(n)}) \\ \gamma=\partial s,\,s\in \varDelta^{n}([3]^{*n+1}),\,a_{0}^{0}\in s}}
\bigg(\sum_{s'\in \varDelta^{n}(\gamma')}l(\hat{f}(s),\hat{f}(s'))\bigg)\nonumber\\
&=& \sum_{\substack{s,\,s'\in \varDelta^{n}([3]^{*n+1}) \\ s\cap s'=\varnothing}}
l(\hat{f}(s),\hat{f}(s'))\nonumber\\
&\equiv& 1\pmod{2}. \nonumber
\end{eqnarray*}
Thus the result follows.

Therefore, $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ are simplicial $n$-complexes that are `intrinsically linked' with respect to embeddings into ${\mathbb R}^{2n}$. We note that the simplicial $n$-complex $\sigma_{2n+2}^{n}\setminus\{|a_{0}a_{1}\cdots a_{n}|\}$ has already been shown to satisfy this property (Freedman--Krushkal \cite[Proposition 4.1]{FK14}; see also Freedman--Krushkal--Teichner \cite[Lemma 6]{FKT94}). However, both our $N_{1}^{(n)}$ and $N_{3}^{(n)}$ are proper subcomplexes of this complex. Moreover, the complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ are minimal in the following sense among all simplicial $n$-complexes possessing this property.

\begin{Proposition}\label{min}
Let $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\to {\mathbb R}^{2n}$ such that ${\rm lk}_{2}(f(\lambda))= 0$ for any element $\lambda$ in $\Lambda^{n-1,n}(N)$.
\end{Proposition}

A central role in the proof of Theorem \ref{ILR2n} is played by the van Kampen--Flores theorem. In the next section, we recall this theorem and present proofs of Theorem \ref{ILR2n} and Proposition \ref{min} using it. The van Kampen--Flores theorem is also used to construct examples of simplicial $n$-complexes that are intrinsically linked with respect to embeddings into $\mathbb{R}^{2n+1}$, see Nikkuni \cite{N26}.

\begin{Remark}\label{nsp}
A graph $G$ is said to be {\it non-separating planar} if there exists an embedding $f: G \to \mathbb{R}^2$ such that, for every cycle $\gamma$ in $G$, all vertices of $G$ not lying on $\gamma$ belong to the same connected component of $\mathbb{R}^2 \setminus f(\gamma)$. Equivalently, $G$ is not non-separating planar if and only if for every embedding $f:G\to \mathbb{R}^2$, the image $f(G)$ contains a two-component link consisting of a $0$-sphere and a $1$-sphere with $\mathbb{Z}_2$-linking number $1$. Dehkordi--Farr proved in \cite{DF} that a graph is non-separating planar if and only if it contains no subgraph homeomorphic to $K_1 \sqcup K_4$, $K_1 \sqcup K_{2,3}$, or $K_{1,1,3}$ as illustrated in Fig. \ref{nonsp}. They further applied this characterization to the study of linkless embeddings of graphs in $\mathbb{R}^3$. For related results, see Fowler--Li--Pavelescu \cite{FLP23}, Pavelescu--Pavelescu \cite{PP24}. Note that $N_1^{(1)}$ is isomorphic to $K_1 \sqcup K_4$, $N_2^{(1)}$ is isomorphic to $K_1 \sqcup K_{2,3}$, and $N_3^{(1)}$ is isomorphic to $K_{1,1,3}$. Thus, our simplicial $n$-complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ may be regarded as higher-dimensional analogues of the minimal subgraphs that prevent a graph from being non-separating planar.
\end{Remark}

\begin{figure}[htbp]
\begin{center}
\scalebox{0.525}{\includegraphics*{nonsp}}
\caption{$K_1 \sqcup K_4$, $K_1 \sqcup K_{2,3}$ and $K_{1,1,3}$}
\label{nonsp}
\end{center}
\end{figure}

A proof sketch is not given here beyond indicating the main tool: “A central role in the proof of Theorem \ref{ILR2n} is played by the van Kampen--Flores theorem.” The authors say that “In the next section, we recall this theorem and present proofs of Theorem \ref{ILR2n} and Proposition \ref{min} using it.”

A proof sketch is not given here beyond indicating the main tool: “A central role in the proof of the main theorem is played by the van Kampen--Flores theorem.” The authors say that “Next we recall this theorem and present proofs of the main theorem and \begin{Proposition}\label{min}
Let $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\to {\mathbb R}^{2n}$ such that ${\rm lk}_{2}(f(\lambda))= 0$ for any element $\lambda$ in $\Lambda^{n-1,n}(N)$.
\end{Proposition} using it.”

Question:
Let $n$ be a positive integer. Write $\sigma_m=|a_0a_1\cdots a_m|$ for an $m$-simplex, and let $\sigma_m^n$ denote its $n$-skeleton. Define three simplicial $n$-complexes by
\[
N_1^{(n)}=\sigma_{2n+2}^n\setminus\{\,|a_0a_{j_1}\cdots a_{j_n}|\mid 1\le j_1<\cdots<j_n\le 2n+2\,\},
\]
so $N_1^{(n)}$ is obtained from $\sigma_{2n+2}^n$ by removing all $n$-simplices containing $a_0$,
\[
N_2^{(n)}=[3]^{*n+1}\setminus\{\,|a_0^0a_{j_1}^1\cdots a_{j_n}^n|\mid j_1,\dots,j_n\in\{0,1,2\}\,\},
\]
where $[3]^{*n+1}=\{a_0^0,a_1^0,a_2^0\}*\{a_0^1,a_1^1,a_2^1\}*\cdots*\{a_0^n,a_1^n,a_2^n\}$ is the simplicial join of $n+1$ three-point sets, and
\[
N_3^{(n)}=\sigma_{2n+2}^n\setminus \Delta^n(\sigma_{n+1}),
\]
so $N_3^{(n)}$ is obtained from $\sigma_{2n+2}^n$ by removing all $n$-simplices contained in the face $\sigma_{n+1}=|a_0a_1\cdots a_{n+1}|$.

For a simplicial complex $K$, let $\Lambda^{n-1,n}(K)$ be the set of unordered pairs $\lambda=\gamma\sqcup\gamma'$ of mutually disjoint subcomplexes of $K$ such that $\gamma$ is homeomorphic to $S^{n-1}$ and $\gamma'$ is homeomorphic to $S^n$. If $f:K\to\mathbb{R}^{2n}$ is an embedding, then $f(\lambda)=f(\gamma)\sqcup f(\gamma')$ is a two-component link in $\mathbb{R}^{2n}$, and ${\rm lk}_2(f(\lambda))\in\mathbb Z_2$ denotes its mod-2 linking number.

Suppose $K$ is one of $N_1^{(n)}$, $N_2^{(n)}$, or $N_3^{(n)}$. Which conclusion about embeddings $f:K\to\mathbb{R}^{2n}$ holds?

For every embedding $f:K\to\mathbb{R}^{2n}$, there exists some $\lambda\in\Lambda^{n-1,n}(K)$ such that ${\rm lk}_2(f(\lambda))=1$; equivalently, $f(K)$ contains disjoint embedded spheres of dimensions $n-1$ and $n$ whose mod-2 linking number is $1$.

Choices:
B. For every embedding $f:K\to\mathbb{R}^{2n}$, one has ${\rm lk}_2(f(\lambda))=1$ for every $\lambda\in\Lambda^{n-1,n}(K)$; equivalently, every pair of disjoint subcomplexes of types $S^{n-1}$ and $S^n$ occurring in $K$ forms a mod-2 linked pair in $\mathbb{R}^{2n}$.
C. For every embedding $f:K\to\mathbb{R}^{2n}$, there exists some $\lambda\in\Lambda^{n-1,n}(K)$ such that ${\rm lk}_2(f(\lambda))\neq 0$.
D. There exists an embedding $f:K\to\mathbb{R}^{2n}$ such that for every $\lambda\in\Lambda^{n-1,n}(K)$, one has ${\rm lk}_2(f(\lambda))=1$; equivalently, each of $N_1^{(n)},N_2^{(n)},N_3^{(n)}$ admits a realization in $\mathbb{R}^{2n}$ in which all disjoint $S^{n-1}\sqcup S^n$ pairs are mod-2 linked.
E. For every embedding $f:K\to\mathbb{R}^{2n}$, there exist disjoint subcomplexes $\lambda=\gamma\sqcup\gamma'\in\Lambda^{n,n}(K)$ such that ${\rm lk}_2(f(\lambda))=1$; equivalently, $f(K)$ always contains a two-component link of two embedded $n$-spheres with mod-2 linking number $1$.

replace explicit value 1 in \mathbb Z_2 by nonzero

quantifier-order on embedding versus linked pairs

sphere-dimension type (n-1,n) replaced by (n,n)

The stem does not explicitly reveal the correct conclusion. It defines the complexes and the mod-2 linking setup, but does not directly signal which quantifier pattern or sphere-dimension pattern is correct.

The item is not a pure restatement: the options vary meaningfully by quantifiers, universality versus existence, and sphere dimensions. A student must distinguish among competing conclusions rather than recognize a verbatim theorem statement.

There is some reasoning pressure because the choices differ by subtle logical strength. However, this is weakened substantially because choice C is effectively equivalent to A: since the linking number is stated to lie in Z_2, saying it is '!= 0' is the same as saying it equals 1.

The distractors are fatally compromised by choice C, which is not genuinely false or weaker in this mod-2 setting and duplicates the intended correct answer. Although B, D, and E are plausible mathematical traps, the A/C ambiguity makes the distractor set poor overall.

Conceptually sophisticated and mostly free of stem-level leakage, but the MCQ is flawed because A and C are both correct in Z_2. This ambiguity seriously weakens its validity as a multiple-choice item.