\label{thm:MinimalSurfaceExistence}
For every $p, q \geq 1$, there exists a minimal embedding of $\mathbb{S}^p \times \mathbb{S}^q \times \mathbb{S}^1$ in $\mathbb{S}^{p+q+2}$.

\begin{theorem}\label{thm:MinimalSurfaceExistence}
For every $p, q \geq 1$, there exists a minimal embedding of $\bS^p \times \bS^q \times \bS^1$ in $\bS^{p+q+2}$.
\end{theorem}

\begin{lemma}\label{lemma:change-of-variables}
For given $(n,k)$, the function $f(t)$ solves equation~\eqref{eqn:capillary-ODE} if and only if the function $f (\sqrt{1-t^2})$ solves this equation for $(n,n-k)$.
Any capillary cones produced by solving this equation with the appropriate free boundary are isometric via the involution $t \leftrightsquigarrow \sqrt{1-t^2}$; in particular, they form the same contact angle.
\end{lemma}

\begin{lemma}\label{lemma:bound-at-sqrt}
Any solution $f$ of equation~\eqref{eqn:capillary-ODE} with two zeros in $(0,1)$ satisfies $f(\sqrt{\frac{k-1}{n-2}}) \leq \frac{2}{\sqrt{n}} \lambda^{- \frac{1}{2}}$.
\end{lemma}

\begin{theorem}\label{thm:capillaryConeExistence}
    For all $n \geq 4$, $2 \leq k \leq n-2$, and $\theta \in (0,\tfrac{\pi}{2}]$, there exists a capillary minimal cone $\tilde{\mathbf{C}}_{n,k,\theta}$ that is topologically $C(\bS^{n-k-1}\times \bS^{k-1}\times [0,1])$ with isometry group $O(n-k) \times O(k)$.

    The double of the free-boundary cone $\tilde{\mathbf{C}}_{n,k,\frac{\pi}{2}}$ is a non-isoparametric minimal hypercone $\hat{\mathbf{C}}_{n,k} \subset \bR^{n+1}$.
    Its link $\mathbf{S}_{n,k} \subset \bS^n$ is topologically $\bS^{n-k-1} \times \bS^{k-1} \times \bS^1$, invariant under $O(n-k) \times O(k) \times \bZ_2$.
\end{theorem}

We construct new minimal surfaces in the sphere as minimal embeddings $\mathbb{S}^p \times \mathbb{S}^q \times \mathbb{S}^1 \hookrightarrow \mathbb{S}^{p+q+2}$, for any $p,q \geq 1$, by doubling the links of free-boundary minimal cones in $\mathbb{R}^{n+1}_+$.
These free-boundary surfaces arise as limits of families of capillary minimal surfaces constructed by solving a free boundary ODE.
Under suitable rescalings these cones converge, in the shallow-angle regime, to a homogeneous solution of the one-phase Bernoulli problem.
Unlike the examples in~\cite{FTW-1}, our families of free-boundary cones do not enjoy additional reflection symmetries and exhibit different geometric properties.
Topologically, the links are diffeomorphic to $\mathbb{S}^{n-k-1}\times \mathbb{S}^{k-1}\times \mathbb{S}^1$ with isometry group $O(n-k)\times O(k) \times \bZ_2$, and the induced metric can be expressed as a doubly warped product over the $\mathbb{S}^1$ factor.
This construction resolves problems posed by Hsiang-Lawson~\cite{hsiang-lawson-jdg}*{Ch.~III \S~3} and Hsiang-Hsiang~\cite{hsiang-hsiang}*{Problem~3} on the existence of minimal submanifolds in the sphere, and advances the Hsiang-Lawson program in low cohomogeneity through a new, self-contained approach.
Our construction also generalizes results of Carlotto-Schulz and Wang-Wang-Zhou~\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to arbitrary pairs of spherical factors.

The construction of minimal hypersurfaces in spheres $\mathbb{S}^n$ has received great attention, leading to the development of many key techniques in differential geometry.
These include the Hsiang-Lawson framework for the study of minimal submanifolds with large symmetry groups~\cites{hsiang-lawson-jdg , spherical-bernstein, spherical-bernstein-2 , ferus-karcher , free-boundary-mcgrath }, Kapouleas' gluing and desingularization methods~\cites{doubling-clifford-torus, doubling-two-sphere, kapouleas-mcgrath-cpam , kapouleas-wiygul-annalen , choe-soret }, min-max constructions~\cites{ willmore-conjecture , infinitely-many-MN , marques-neves-song , haslhofer-ketover , XinZhouAnnals, WangZhouFourSpheres, song , ko-min-max , spherical-bernstein-xin-zhou }, and the Allen-Cahn approach~\cites{guaraco-phase-transitions , allen-cahn-3-manifolds}.
Separately, White~\cite{white-mapping-degree} constructed complete minimal surfaces asymptotic to a given minimizing cone $\mathbf{C}$ provided that its link is not a homology sphere.
The topological complexity of the links of cones is connected to their density, as proved by Ilmanen-White and Bernstein-Wang~\cites{ilmanen-white, bernstein-wang} using techniques from mean curvature flow.

Our first result presents the following:

Our first result presents the following:

Theorem~\ref{thm:MinimalSurfaceExistence} is obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem, first introduced in~\cite{FTW-1}.

\begin{theorem}\label{thm:capillaryConeExistence}
 For all $n \geq 4$, $2 \leq k \leq n-2$, and $\theta \in (0,\tfrac{\pi}{2}]$, there exists a capillary minimal cone $\tilde{\mathbf{C}}_{n,k,\theta}$ that is topologically $C(\bS^{n-k-1}\times \bS^{k-1}\times [0,1])$ with isometry group $O(n-k) \times O(k)$.

As a consequence of our general technique, we extend the results of~\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to the capillary setting as follows.
\begin{theorem}\label{thm:carlotto-schulz}
 For every $k \geq 2$, there exists a smooth one-parameter family of capillary cones $\{ \bar{\mathbf{C}}_{k,a} \}_{a \in (0,a^*_k] } \subset \bR^{2k+1}$ that interpolates between a free-boundary minimal cone and a homogeneous solution of the one-phase problem through cones of every angle.
 More precisely,
 \begin{enumerate}[(i)]
 \item As $a \downarrow 0$, the rescaled cones $\frac{1}{a} \bar{\mathbf{C}}_{k,a}$ converge in $C^{\infty}_{\textup{loc}}$ to the graph of a homogeneous solution of the one-phase problem~\eqref{eqn:one-phase-problem}.
 \item As $a \in (0,a^*_k]$, the cones $\bar{\mathbf{C}}_{k,a}$ attain every capillary angle $\theta \in (0,\frac{\pi}{2}]$.
 \item The cone $\bar{\mathbf{C}}_{k,a^*_k}$ is the free-boundary cone $\hat{\mathbf{C}}_{2k,k,\frac{\pi}{2}}$, whose double has link given by a minimal embedded $\bS^{k-1} \times \bS^{k-1} \times \bS^1 \hookrightarrow \bS^{2k}$.
 \end{enumerate}
\end{theorem}
This theorem provides further evidence for the importance of the capillary functional in the study of minimal submanifolds, as it provides a bridge between minimal surfaces and solutions of the one-phase problem~\eqref{eqn:one-phase-problem}.
The first instance of such an interpolating family was discovered in~\cite{FTW-1}.

\section{Construction of capillary surfaces}\label{section:construction-capillary}
We adopt many conventions from~\cite{FTW-1}.
Let $n \geq 4$ and $2 \leq k \leq n-2$, and decompose $\bR^{n+1} = \bR^{n-k}_x \oplus \bR^{k}_y \oplus \bR_z$.
For the capillary problem, we consider the upper half-space $\bR^{n+1}_+$ with boundary $\Pi := \{z = 0\} = \partial \{z \geq 0 \}$, so we assume $z \geq 0$ throughout.
The $O(n-k)\times O(k)$-action on the $(x,y)$-variables reduces the minimal surface equation to an ODE, where a natural equivariant coordinate system for the analysis is $\rho := \sqrt{|x|^2 + |y|^2}$ and $t := \frac{|y|}{\sqrt{|x|^2 + |y|^2}}$.
As computed in~\cite{FTW-1}, a graphical surface $\mathbf{C} = \{ z = U(|x|,|y|) > 0\}$ is a minimal cone if and only if $U = \rho f(t)$, where $f(t)$ satisfies the ODE
\begin{equation}\label{eqn:ode-star}\tag{$\star$}
 (1-t^2) f'' + (f - tf') + (n-2) \left( 1 + (1-t^2) \frac{(f')^2}{1 + f^2} \right) (f - A_{n,k} f') = 0,
\end{equation}
where $A_{n,k}(t) := t - \frac{k-1}{n-2} t^{-1}$.
The boundary angle for $\mathbf{C}$ at a point $t_*$ where $f(t_*) = 0$ is given by
\begin{equation}\label{eqn:boundaryAngleTerm}
 \theta = \on{arctan} (\sqrt{1 - t_*^2} \, |f'(t_*)|)
\end{equation}
A useful phenomenon in the study of families of capillary solutions is that, as the contact angle $\theta \downarrow 0$, the rescaled functions $\frac{1}{\tan \theta} f_{\theta}$ converge (subsequentially) to a multiple of a solution of the one-phase Bernoulli problem; this idea was utilized in~\cites{improved-regularity,FTW-1}.
For $\lambda \in (0,\infty)$ and $f$ a solution of~\eqref{eqn:ode-star}, the rescaled function $\frac{1}{\sqrt{\lambda}} f$ is a solution of the equation with parameter
\begin{equation}\label{eqn:capillary-ODE} \tag{$\star_\lambda$}
 (1-t^2) f'' + (f - tf') + (n-2) \left( 1 + (1-t^2) \frac{\lambda (f')^2}{1 + \lambda f^2} \right) (f - A_{n,k} f') = 0.
\end{equation}
Motivated by the above limiting behavior, we will study equation~\eqref{eqn:capillary-ODE} uniformly in $\lambda \in [0,\infty]$.
Then, equation~\eqref{eqn:capillary-ODE} includes the one-phase problem in the limit $\lambda \downarrow 0$; the resulting homogeneous solutions are further analyzed in~\cites{FTW-1 , FTW-stability-one-phase , six-way}.
When $\lambda = 0$, the problem~\eqref{eqn:capillary-ODE} specializes to the hypergeometric Legendre operator associated to $p(t) := t^{k-1} (1-t^2)^{\frac{n-k}{2}}$,
\begin{equation}\label{eqn:legendre-form}
\cL_{n,k} f = (1-t^2) f'' + (n-1) (f - t f') + (k-1) t^{-1} f' = \tfrac{1-t^2}{p(t)} \left[ (p f')' + (n-1) \tfrac{p}{1-t^2} f \right].
\end{equation}
The solutions of $\cL_{n,k} f_0 = 0$ have the significance that $U(x,y) := \rho f_0(t)$ is harmonic in $\bR^n$, while the analogue of the boundary condition~\eqref{eqn:boundaryAngleTerm} implies that $|\nabla U| =1$ along $\partial \{ U > 0 \}$.
Consequently, $U$ is a solution of the \textit{one-phase Bernoulli} problem
\begin{equation}\label{eqn:one-phase-problem}\tag{OP}
 \Delta u = 0 \; \text{ in } \; \{ u > 0 \}
 \qquad \text{and}\qquad
 |\nabla u| = 1 \; \text{ on } \; \partial \{ u > 0 \}.
\end{equation}
This problem is closely connected to the theory of minimal surfaces, with many constructions and theorems having direct counterparts.
In low dimensions, classical methods such as the Weierstrass representation and gluing techniques have led to classification results and new examples~\cites{ entire-hairpins, traizet , jerison-kamburov, n-dim-catenoid , hines-kolesar-mcgrath}.
We refer the reader to~\cites{ one-phase-simon-solomon , FTW-1 , FTW-stability-one-phase } for some recent results on the one-phase problem and its connections to the theory of minimal surfaces.
As in~\cite{FTW-1}, Theorems~\ref{thm:capillaryConeExistence} and~\ref{thm:carlotto-schulz} deepen these connections by bridging minimal surfaces and one-phase solutions through interpolating families of capillary surfaces.

The capillary minimal cone $\mathbf{C} = \text{graph}(\rho f(t))$ admits the spherical parametrization
\[
F(\rho, t , \xi , \eta) = \left( \rho \sqrt{1-t^2} \xi, \rho t \eta , \rho f(t)\right) , \qquad \xi \in \bS^{n-k-1}, \; \eta \in \bS^{k-1},
\]
for which $|F(\rho, t , \xi, \eta)|^2 = \rho^2 ( 1+ f(t)^2)$.
Consequently, the link of the capillary cone is a $O(n-k) \times O(k)$-invariant surface $\Sigma \subset \bS^n_+$ inside the upper hemisphere with parametrization
\begin{equation}\label{eqn:link-equation}
 \Sigma := \left\{ \left( \frac{\sqrt{1-t^2}}{\sqrt{1+f^2}} \xi, \frac{t}{\sqrt{1+f^2}} \eta, \frac{f}{\sqrt{1+f^2}} \right) : (\xi, \eta, t) \in \bS^{n-k-1} \times \bS^{k-1}\times I_\theta \right\} \subset \bS^n_+
\end{equation}
where $I_{\theta} \subset [0,1]$ denotes the positive phase of $f$.
For cones of type $(ii)$, the resulting function $U = \rho f(t)$ corresponds to a graph over a conical annulus in the base, of the form
\[
\Gamma := \{ (x,y) : t_1 \rho < |y| < t_2 \rho \}
\]
with two boundary components on $\Pi = \{ p_{n+1} \geq 0 \}$, namely the two cones $\{ |y| = t_i \rho \}$.
Topologically, the link of such a cone is diffeomorphic to $\bS^{n-k-1} \times \bS^{k-1} \times I_\theta$.
We denote such a capillary cone with contact angle $\theta$ by $\tilde{\mathbf{C}}_{n,k,\theta}$, in parallel with the cones $\mathbf{C}_{n,k,\theta}$ of type $(i)$, constructed in~\cite{FTW-1}*{Theorem 1.3}.
By construction, $\tilde{\mathbf{C}}_{n,k,\theta}$ is a regular capillary cone with an isolated singularity at the origin.

Theorem~\ref{thm:MinimalSurfaceExistence} is obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem, first introduced in~\cite{FTW-1}.

\begin{theorem}\label{thm:capillaryConeExistence}
    For all $n \geq 4$, $2 \leq k \leq n-2$, and $\theta \in (0,\tfrac{\pi}{2}]$, there exists a capillary minimal cone $\tilde{\mathbf{C}}_{n,k,\theta}$ that is topologically $C(\mathbb{S}^{n-k-1}\times \mathbb{S}^{k-1}\times [0,1])$ with isometry group $O(n-k) \times O(k)$.

    The double of the free-boundary cone $\tilde{\mathbf{C}}_{n,k,\frac{\pi}{2}}$ is a non-isoparametric minimal hypercone $\hat{\mathbf{C}}_{n,k} \subset \mathbb{R}^{n+1}$.
    Its link $\mathbf{S}_{n,k} \subset \mathbb{S}^n$ is topologically $\mathbb{S}^{n-k-1} \times \mathbb{S}^{k-1} \times \mathbb{S}^1$, invariant under $O(n-k) \times O(k) \times \bZ_2$.
\end{theorem}
The resulting minimal submanifolds of $\mathbb{S}^n$ are non-isoparametric, meaning that they have non-constant principal curvatures.
Isoparametric minimal surfaces in the sphere have been classified using algebraic and analytic techniques, cf.~\cites{cecil-chi-jensen, miyaoka, chi-four-curvatures}.
Theorem~\ref{thm:MinimalSurfaceExistence} is also connected to questions posed by Choe-Fraser~\cite{choe-fraser}*{\S~4} on the possible topology and diffeomorphism types of minimal submanifolds inside manifolds with positive Ricci curvature.

A minimal surface $\Sigma \subset \mathbb{S}^n$ with reflection symmetry $p_{n+1} \leftrightsquigarrow -p_{n+1}$, restricted to the upper half-space $\mathbb{R}^{n+1}_+$ and $\mathbb{S}^n_+$, forms a free-boundary surface.
Conversely, since the boundary of $\mathbb{S}^n_+$ is totally geodesic, any free-boundary surface can be doubled to produce an embedded minimal surface in the sphere.
Therefore, when viewed as a graph, any such free-boundary minimal surface must have infinite slope along the boundary equator, making the graphical shooting approach singular in these coordinates.
Using a bi-orthogonal symmetry, the capillary minimal surface equation reduces to a free boundary ODE.
For capillary angles $\theta \in (0,\tfrac{\pi}{2})$, shooting techniques that match the contact angle are well-defined, and we show that the resulting solutions converge uniformly to an infinite-slope shot.
This yields the free-boundary minimal surface in Theorem~\ref{thm:MinimalSurfaceExistence}.

The most symmetric case of Theorem~\ref{thm:MinimalSurfaceExistence} where $n=2k$ (hence $n-k-1=k-1$), recovers the minimal hypertori produced by Carlotto-Schulz~\cite{carlotto-schulz}.
Using numerical methods, Perdomo approximated the spectrum and index of the Carlotto-Schulz examples~\cites{perdomo-spectrum, perdomo-carlotto-schulz}.
Analogues of this construction for constant mean curvature hypersurfaces are discussed in~\cites{huang-wei-2022 , perdomo-navigating}.
Very recently, Wang-Wang-Zhou~\cite{spherical-bernstein-xin-zhou} provided an alternative construction of embedded spheres $\mathbb{S}^3$ and hypertori $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ inside the $4$-sphere, using equivariant min-max theory.

As a consequence of our general technique, we extend the results of~\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to the capillary setting as follows.
\begin{theorem}\label{thm:carlotto-schulz}
    For every $k \geq 2$, there exists a smooth one-parameter family of capillary cones $\{ \bar{\mathbf{C}}_{k,a} \}_{a \in (0,a^*_k] } \subset \mathbb{R}^{2k+1}$ that interpolates between a free-boundary minimal cone and a homogeneous solution of the one-phase problem through cones of every angle.
    More precisely,
    \begin{enumerate}[(i)]
        \item As $a \downarrow 0$, the rescaled cones $\frac{1}{a} \bar{\mathbf{C}}_{k,a}$ converge in $C^{\infty}_{\textup{loc}}$ to the graph of a homogeneous solution of the one-phase problem~\eqref{eqn:one-phase-problem}.
        \item As $a \in (0,a^*_k]$, the cones $\bar{\mathbf{C}}_{k,a}$ attain every capillary angle $\theta \in (0,\frac{\pi}{2}]$.
        \item The cone $\bar{\mathbf{C}}_{k,a^*_k}$ is the free-boundary cone $\hat{\mathbf{C}}_{2k,k,\frac{\pi}{2}}$, whose double has link given by a minimal embedded $\mathbb{S}^{k-1} \times \mathbb{S}^{k-1} \times \mathbb{S}^1 \hookrightarrow \mathbb{S}^{2k}$.
    \end{enumerate}
\end{theorem}
This theorem provides further evidence for the importance of the capillary functional in the study of minimal submanifolds, as it provides a bridge between minimal surfaces and solutions of the one-phase problem~\eqref{eqn:one-phase-problem}.
The first instance of such an interpolating family was discovered in~\cite{FTW-1}.

The general case of $n \neq 2k$ in Theorem~\ref{thm:capillaryConeExistence} (corresponding to $p \neq q$ in Theorem~\ref{thm:MinimalSurfaceExistence}) is more challenging than the case $n=2k$ ($p=q$), which enjoys an additional symmetry.
In the low cohomogeneity framework of Hsiang, Hsiang, and Lawson~\cites{hsiang-lawson-jdg , hsiang-hsiang }, the existence of solutions with certain symmetries was obtained via the study of planar dynamical systems; however, their technique encounters difficulties in the current setting, as discussed in~\cite{carlotto-schulz}.
Carlotto-Schulz obtain their surfaces by studying a non-planar, $3 \times 3$ ODE system.
The shooting method was also used in Angenent's construction of self-shrinking tori~\cite{angenent-tori}.

The main part of our paper is devoted to developing a novel, singular shooting method for graphical solutions of the capillary problem, allowing us to study the minimal surfaces and cones in question via single-variable ODE methods.
Notably, this provides an effective method to approximate these surfaces, which is often inaccessible through pure variational methods.
The existence of a minimally embedded $\mathbb{S}^{k-1} \times \mathbb{S}^{k-1} \times \mathbb{S}^1 \hookrightarrow \mathbb{S}^{2k}$, connected to a homogeneous one-phase solution by a smooth interpolating family of capillary hypersurfaces, follows using only Lemma~\ref{lemma:change-of-variables} (symmetry under $t \leftrightsquigarrow \sqrt{1-t^2}$) and Lemma~\ref{lemma:bound-at-sqrt} (a uniform bound for shots reaching zero) to prove Theorem~\ref{thm:carlotto-schulz}.
The constructions of Theorems~\ref{thm:MinimalSurfaceExistence} and~\ref{thm:capillaryConeExistence} are also connected to new $O(n-k) \times O(k)$-invariant homogeneous one-phase solutions constructed and studied in a related paper~\cite{six-way}.

Theorem~\ref{thm:MinimalSurfaceExistence} is described as being “obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem.” The introduction explains the proof strategy via a capillary-to-free-boundary limiting argument: using “a bi-orthogonal symmetry, the capillary minimal surface equation reduces to a free boundary ODE.” For capillary angles $\theta\in(0,\tfrac{\pi}{2})$, “shooting techniques that match the contact angle are well-defined,” and the authors “show that the resulting solutions converge uniformly to an infinite-slope shot.” Since doubling across the totally geodesic boundary of $\mathbb{S}^n_+$ turns a free-boundary surface into an embedded minimal surface in the sphere, this limiting “infinite-slope shot” “yields the free-boundary minimal surface in Theorem~\ref{thm:MinimalSurfaceExistence}.”

Theorem~\ref{thm:MinimalSurfaceExistence} is described as being “obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem.” The introduction explains the proof strategy via a capillary-to-free-boundary limiting argument: using “a bi-orthogonal symmetry, the capillary minimal surface equation reduces to a free boundary ODE.” For capillary angles $\theta\in(0,\tfrac{\pi}{2})$, “shooting techniques that match the contact angle are well-defined,” and the authors “show that the resulting solutions converge uniformly to an infinite-slope shot.” Since doubling across the totally geodesic boundary of $\mathbb{S}^n_+$ turns a free-boundary surface into an embedded minimal surface in the sphere, this limiting “infinite-slope shot” is used in establishing the main theorem.

Question: Let $p,q\ge 1$ be integers. Here $\mathbb S^m$ denotes the standard unit sphere, and a minimal embedding of a manifold into $\mathbb S^N$ means a smooth embedding whose image is a minimal submanifold of $\mathbb S^N$. Which existence statement holds for these parameters?

For every $p,q\ge 1$, there exists a minimal embedding $\mathbb S^p\times \mathbb S^q\times \mathbb S^1\hookrightarrow \mathbb S^{p+q+2}$.

Choices:
B. For every $p,q\ge 1$, there exists a minimal embedding $\mathbb S^p\times \mathbb S^q\times \mathbb S^1\hookrightarrow \mathbb S^{p+q+1}$.
C. For every $p,q\ge 1$, there exists a smooth embedding $\mathbb S^p\times \mathbb S^q\times \mathbb S^1\hookrightarrow \mathbb S^{p+q+2}$.
D. For every $p,q\ge 1$, there exists an immersed minimal submanifold $\mathbb S^p\times \mathbb S^q\times \mathbb S^1\looparrowright \mathbb S^{p+q+2}$.
E. There exists an integer $N$ such that for every $p,q\ge 1$, there exists a minimal embedding $\mathbb S^p\times \mathbb S^q\times \mathbb S^1\hookrightarrow \mathbb S^{N}$.


\label{thm:isopery}
    We have $\gamma(\alpha, n) \leqslant \frac{(1-\alpha)(1-(1-\alpha)^n)}{\alpha n}$ for all $n\geqslant 1$ and $\alpha \in (0, 1)$.

\begin{theorem}\label{thm:isopery}
    We have $\gamma(\alpha, n) \le \frac{(1-\alpha)(1-(1-\alpha)^n)}{\alpha n}$ for all $n\ge 1$ and $\alpha \in (0, 1)$.
\end{theorem}

Let $\Omega$ be a finite set and let $n\geqslant 2$. Let $\mu=\mu_n$ be the uniform probability measure on $\Omega^n$. We say that an ordered pair of words $(w, u) \in \Omega^n \times \Omega^n$ {\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \ldots, w_n)$, $u=(u_1, \ldots, u_n)$ then for some $j\in \{1, \ldots, n\}$ we have $(w_{n-j+1}, \ldots, w_n) = (u_1, \ldots, u_j)$. Note that we in particular allow $u = w$.
We are interested in the following extremal question: suppose that $A, B\subset \Omega^n$ are sets of words such that no two words $w \in A$ and $u \in B$ overlap. For what pairs of densities $\alpha, \beta \in (0,1)$ is it possible to have $\mu(A) \geqslant \alpha$ and $\mu(B) \geqslant \beta$?

There is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \subset \Omega^n$ such that no two distinct words in $A$ overlap (see \cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.

Define the shift map $s=s_n: \Omega^n \rightarrow \Omega^{n-1}$ by
\[
s(w_1, \ldots, w_n) = (w_2, \ldots, w_n).
\]
For a subset $A \subset \Omega^n$ we can define the set of words which do not overlap with $A$ as follows:
\[
U= U(A) = \Omega^n \setminus \bigcup_{j=0}^{n-1} s^j(A) \times \Omega^{j}.
\]
It is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\in A$.
Let $\gamma(\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\subset \Omega^n $ of measure $\alpha$ and all finite sets $\Omega$. Here we consider the uniform measure on the space $\Omega^n$.
Note that if $A \subset A'$ then we have the inclusion $U(A') \subset U(A)$. This means that
$\gamma(\alpha, n)$ is a monotone decreasing function in $\alpha$. For example, it is an easy exercise to show that $\gamma(\alpha,2) = \max\{ (1-\alpha)^2, ~1-\alpha^{1/2} \}$ holds for any $\alpha \in (0,1)$.

Our result is the following estimate.

Our result is the following estimate.

So if we have a pair of non-overlapping sets $A, B\subset \Omega^n$ with densities $\alpha$ and $\beta$ then we have $\alpha\beta n\le (1-\alpha)(1-(1-\alpha)^n) \le 1 $.

We can interpolate between these two examples by taking $A = \Omega^{n-k} \times S^k$ for some $1 \le k \le n$.
Then the set $U = U(A)$ is given by
\begin{align*}
U = \{ (w_1, \ldots, w_n):~ w_1 \not\in S, \quad \{w_{j+1}, \ldots, w_{j+k}\}\not\subset S,~j=0, \ldots, n-k \}.
\end{align*}
The exact formula for $\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \cite{arratia1989two} (see also \cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.
Denote $p = |S|/|\Omega|$ so that $\alpha = p^k$.
Let $Z_1, \ldots, Z_{n-1} \sim \operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:
\[
\mu(U) = (1-p)\Pr[ R_{n-1} < k].
\]
Indeed, we can view $w_2, \ldots, w_n \in \Omega$ as iid variables uniformly distributed on $\Omega$ and select $Z_i = 1_{w_{i+1} \in S}$.
By \cite[Example 3]{arratia1989two} we have the following estimate on this probability:
\[
\left|\Pr[ R_{n-1} < k] - e^{-\lambda}\right| \le \frac{\lambda (2k+1)}{n-1} + 2 p^k, \quad \lambda = p^k ((n-2)(1-p)+1)
\]
Let $k = [n\alpha \log(1/\alpha) ]$. Then for $1/n \ll \alpha \ll 1$ we have $p = \alpha^{1/k} = e^{-\frac{\log(1/\alpha)}{k}} = 1 - \frac{1+o(1)}{n\alpha}$. This gives $\lambda = 1+o(1)$ and $\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have
\[
\mu(U) = \frac{e^{-1}+o(1)}{\alpha n},
\]
where $o(1)$ tends to zero as $\min(\alpha^{-1}, n\alpha) \to \infty$. This matches the bound in Theorem \ref{thm:isopery} for all $\alpha \in (1/n, 1/2)$ up to a constant factor.

We trivially have $\beta_{1}=\alpha_{1}$. Since $A_{j} = s(A_{j+1})$, we have the inclusion $A_{j+1} \subset \Omega\times A_{j} $. We have $B_{j+1} = (B_j \times \Omega) \cup A_{j+1}$ and, in particular $B_j \times \Omega \subset B_{j+1}$. Together these observations imply the following chain of inequalities:
\[
\beta_n \ge \ldots \ge \beta_1 = \alpha_1 \ge \ldots \ge \alpha_n.
\]
Now let us define sets $D_j$ as follows:
\[
D_j = A_j \setminus (B_{j-1} \times \Omega) = B_j \setminus (B_{j-1} \times \Omega),
\]
where for $j=1$ we put $D_1 = A_1=B_1$.
In particular, since $B_{j-1} \times \Omega \subset B_{j}$, we can write $B_j$ as a disjoint union $(B_{j-1}\times \Omega) \sqcup D_j$ and
$\mu(D_j) = \beta_j-\beta_{j-1}$ for all $j=1, \ldots, n$ (where we set $\beta_0=0$).
Note that we can write
\begin{align*}
B_j &= D_j \sqcup (B_{j-1}\times \Omega) = D_j \sqcup (D_{j-1}\times \Omega) \sqcup (B_{j-2}\times \Omega^2) = \ldots \\
&= D_j \sqcup (D_{j-1}\times \Omega) \sqcup (D_{j-2}\times \Omega^2) \sqcup \ldots \sqcup (D_{1}\times \Omega^{j-1}) \\
& = A_j \cup \left( (D_{j-1}\times \Omega) \sqcup (D_{j-2}\times \Omega^2) \sqcup \ldots \sqcup (D_{1}\times \Omega^{j-1}) \right).
\end{align*}
Using Observation \ref{obs1}, we have the following bounds for $i=1, \ldots, j-1$:
\[
\mu(A_j \cap (D_i \times \Omega^{j-i})) \le \lambda_{j, i}\cdot \mu(D_i) \le \alpha_{j-i} \cdot \mu(D_i)
\]
So since sets $D_i \times \Omega^{j-i}$ are pairwise disjoint, we obtain
\[
\mu(B_j) = \mu(A_j) + \sum_{i=1}^{j-1} \mu(D_i \times \Omega^{j-i} \setminus A_j) \ge \mu(A_j) + \sum_{i=1}^{j-1} (1- \alpha_{j- i}) \mu(D_{i})
\]
giving the following relation between $\alpha$-s and $\beta$-s:
\begin{equation}\label{eq:beta}
 \beta_j \ge \alpha_j + \sum_{i=1}^{j-1} (1-\alpha_{j-i}) (\beta_{i}-\beta_{i-1}).
\end{equation}
Denote $\gamma_i = 1-\beta_i = \mu(U(A_i))$ and let $\delta_i = \alpha_{i-1}-\alpha_i$ for $i=1, \ldots, n$ where we put $\alpha_0=1$ and $\gamma_0=1$. Then (\ref{eq:beta}) can be rewritten as follows:
\begin{equation}\label{eq:gamma}
 \gamma_j \le \sum_{i=0}^{j-1} \gamma_i\delta_{j-i}.
\end{equation}
We also have the following information about $\gamma_i, \delta_i$:
\[
\gamma_n \le \gamma_{n-1} \le \ldots \le \gamma_1 \le \gamma_0 = 1,
\]
\[
\delta_1+\ldots+\delta_n = 1-\alpha, \quad \delta_i \ge 0, \quad i=1, \ldots, n.
\]
We will use these properties to upper bound $\gamma_n$. The idea is to use (\ref{eq:gamma}) to compare $\gamma$ with a random walk on $\Z$.

Let $p_i = \frac{\delta_i}{1-\alpha}$ and let $Z$ be the random variable supported on $\{1, \ldots, n\}$ given by the following distribution:
\[
\Pr[Z = i] = p_i.
\]
Let $Z_1, \ldots, Z_n$ be i.i.d. copies of $Z$.
\begin{obs}
 For every $j=0, \ldots, n$ we have the following:
 \begin{equation}\label{eq:probability}
 \gamma_j \le \sum_{s= 0}^n (1-\alpha)^s \Pr[Z_1+\ldots+Z_s = j].
 \end{equation}
\end{obs}
\begin{proof}
 We prove this by induction on $j$. The base case $j=0$ is clear since only $s=0$ term contributes. Now for $j\ge 1$ we have by (\ref{eq:gamma}):
 \begin{align*}
 \gamma_j &\le \sum_{i=0}^{j-1} \gamma_i \delta_{j-i} =
 (1-\alpha) \sum_{i=0}^{j-1} \gamma_i p_{j-i} \\
 &\le (1-\alpha) \sum_{i=0}^{j-1} \sum_{s \ge 0} (1-\alpha)^s \Pr[Z_1+\ldots+Z_s = i] \Pr[Z_{s+1} = j-i] \\
 &=\sum_{s\ge 0} (1-\alpha)^s\Pr[Z_1+\ldots+Z_s = j].
 \end{align*}
\end{proof}

\begin{cor}\label{cor:isoperimetry}
 Let $A \subset \Omega^n$ be a set of measure $\alpha \in (0,1)$. For $w \in \Omega^n$ define $f(w)$ to be the number of indices $j\in \{0, \ldots, n-1\}$ such that $w \in s^j(A) \times \Omega^{j}$.
 Then for every integer $t \in [1, n/4]$ we have the following level set estimate on $f$:
 \begin{equation}
 \mu(\{w\in \Omega^n:~ f(w) \le t\}) \le \frac{8t}{\alpha n}.
 \end{equation}
\end{cor}

Let $\tilde n = [n/2t]$ and $r = n-2t \tilde n$. By the assumption on $t$ we have $\tilde n\ge 1$. Consider a new alphabet $\tilde \Omega = \Omega^{2t}$ and
 let $\tilde s = \tilde s_j: \tilde \Omega^{j} \rightarrow \tilde \Omega^{j-1}$ denote the shift map defined on words over the alphabet $\tilde \Omega$. By identifying $\tilde \Omega^j = \Omega^{2t j}$, we get that $\tilde s = s^{2t}$. For $i=0, \ldots, 2t-1$ let
 \[
 \tilde A_i = s^{r+i}(A) \times \Omega^{i} \subset \Omega^{2t \tilde n} = \tilde \Omega^{\tilde n}.
 \]
 Note that $\mu(\tilde A_i) \ge \mu(A) =\alpha$ for $i=0, \ldots, 2t-1$. For an arbitrary subset $\tilde A \subset \tilde \Omega^{\tilde n}$ we denote $\tilde U(\tilde A) = \tilde \Omega^{\tilde n} \setminus \bigcup_{j=0}^{\tilde n-1} \tilde s^j(\tilde A) \times \tilde \Omega^{j}$, that is the analogue of $U(A)$ over the new alphabet.
 Let $w \in \Omega^n$ and denote $\tilde w = s^r(w) \in \tilde\Omega^{\tilde n}$.
 Note that we have $\tilde w \not \in \tilde U(\tilde A_i)$ precisely when there exists $j \in \{0, \ldots, \tilde n-1\}$ such that $\tilde w \in \tilde s^j(\tilde A_i) \times \tilde \Omega^j$. The latter is in turn equivalent to $w \in s^{2t j + i + r}(A)$. It follows that we have
 \[
 \#\{i \in \{0, \ldots, 2t-1\}: \tilde w \not \in \tilde U(\tilde A_i)\} \le \#\{ i\in \{0, \ldots, n-1\}: w \in s^{i}(A)\times \Omega^i \} = f(w)
 \]
 Thus, if $f(w) \le t$ then there are at least $2t-t=t$ indices $i \in \{0, \ldots, 2t-1\}$ such that $\tilde w \in \tilde U(\tilde A_i)$. So by the union bound and Theorem \ref{thm:isopery} applied to each $\tilde U(\tilde A_i)$ we have
 \[
 t\mu(\{w:~f(w) \le t\}) \le \sum_{i=0}^{2t-1} \mu(\tilde U(\tilde A_i)) \le 2t \frac{(1-\alpha) (1 - (1-\alpha)^{\tilde n})}{\alpha \tilde n} \le \frac{2t}{\alpha \tilde n}.
 \]
 So recalling that $\tilde n = [n / 2t]$ we get
 \[
 \mu(\{w:~f(w) \le t\}) \le \frac{2}{\alpha [n/2t]} \le \frac{8t}{\alpha n}
 \]
 provided that $n \ge 4t$, concluding the proof.
\end{proof}

So if we have a pair of non-overlapping sets $A, B\subset \Omega^n$ with densities $\alpha$ and $\beta$ then we have $\alpha\beta  n\leqslant (1-\alpha)(1-(1-\alpha)^n) \leqslant 1 $.

The proof of Theorem \ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\Omega^n \setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece. This then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts. This relationship can be interpreted in terms of a certain random walk leading to the desired estimate.

We close this section by considering some examples essentially matching the upper bound in Theorem \ref{thm:isopery}. Let $S \subset \Omega$ be an arbitrary subset.
Then for $A = S^n$ one can check that
\[
U(A) = (\Omega\setminus S)\times \Omega^{n-1}
\]
and so we get
\[
\mu(U(A)) = 1-\alpha^{1/n} = \frac{\log(1/\alpha)}{n} + O\left(\frac{\log^2(1/\alpha)}{n^2}\right).
\]
This is a good bound for $\alpha \in (1/2, 1)$.

Similarly, for $A = \Omega^{n-1} \times S$ one can check that
\[
U(A) = (\Omega\setminus S)^n
\]
so that we get
\[
\mu(U(A)) = (1-\alpha)^n.
\]
This is a good bound for $\alpha \in (0, 1/n)$.

We can interpolate between these two examples by taking $A = \Omega^{n-k} \times S^k$ for some $1 \leqslant k \leqslant n$.
Then the set $U = U(A)$ is given by
\begin{align*}
U =  \{ (w_1, \ldots, w_n):~ w_1 \not\in S, \quad \{w_{j+1}, \ldots, w_{j+k}\}\not\subset S,~j=0, \ldots, n-k \}.
\end{align*}
The exact formula for $\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \cite{arratia1989two} (see also \cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.
Denote $p = |S|/|\Omega|$ so that $\alpha = p^k$.
Let $Z_1, \ldots, Z_{n-1} \sim \operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:
\[
\mu(U) = (1-p)\Pr[ R_{n-1} < k].
\]
Indeed, we can view $w_2, \ldots, w_n \in \Omega$ as iid variables uniformly distributed on $\Omega$ and select $Z_i = 1_{w_{i+1} \in S}$.
By \cite[Example 3]{arratia1989two} we have the following estimate on this probability:
\[
\left|\Pr[ R_{n-1} < k] - e^{-\lambda}\right| \leqslant \frac{\lambda (2k+1)}{n-1} + 2 p^k, \quad \lambda = p^k ((n-2)(1-p)+1)
\]
Let $k = [n\alpha \log(1/\alpha) ]$. Then for $1/n \ll \alpha \ll 1$ we have $p = \alpha^{1/k} = e^{-\frac{\log(1/\alpha)}{k}} = 1 - \frac{1+o(1)}{n\alpha}$. This gives $\lambda = 1+o(1)$ and $\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have
\[
\mu(U) = \frac{e^{-1}+o(1)}{\alpha n},
\]
where $o(1)$ tends to zero as $\min(\alpha^{-1}, n\alpha) \to \infty$. This matches the bound in Theorem \ref{thm:isopery} for all $\alpha \in (1/n, 1/2)$ up to a constant factor.

The post-theorem introduction says: "The proof of Theorem \ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\Omega^n \setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece." This "then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts." Finally, "[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate."

The post-theorem introduction says: "The proof of the main theorem is presented in the next section. The rough idea is to split the set $\Omega^n \setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece." This "then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts." Finally, "[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate."

Question:
For a finite set $\Omega$ and an integer $n\ge 1$, let $\mu$ be the uniform probability measure on $\Omega^n$. Let $s: \Omega^n\to\Omega^{n-1}$ be the shift map $s(w_1,\dots,w_n)=(w_2,\dots,w_n)$, and let $s^j$ denote its $j$-fold iterate (with $s^0$ the identity). For a set $A\subset \Omega^n$, define
\[
U(A)=\Omega^n\setminus \bigcup_{j=0}^{n-1} s^j(A)\times \Omega^j,
\]
so $U(A)$ is the set of words in $\Omega^n$ that do not overlap any word of $A$. For $\alpha\in(0,1)$, let $\gamma(\alpha,n)$ be the largest possible value of $\mu(U(A))$ over all finite alphabets $\Omega$ and all sets $A\subset \Omega^n$ with $\mu(A)=\alpha$. Which statement holds for every $n\ge 1$ and every $\alpha\in(0,1)$?

$\gamma(\alpha,n)\le \dfrac{(1-\alpha)\bigl(1-(1-\alpha)^n\bigr)}{\alpha n}$.

Choices:
B. $\gamma(\alpha,n)\le \dfrac{(1-\alpha)\bigl(1-(1-\alpha)^{n-1}\bigr)}{\alpha n}$.
C. $\gamma(\alpha,n)\le \dfrac{1-\alpha}{\alpha n}$.
D. $\gamma(\alpha,n)\le \dfrac{(1-\alpha)\bigl(1-(1-\alpha)^n\bigr)}{\alpha}$.
E. $\gamma(\alpha,n)= \dfrac{(1-\alpha)\bigl(1-(1-\alpha)^n\bigr)}{\alpha n}$.


Question:
Let $n\ge 1$ and let $\alpha\in(0,1)$. For a finite alphabet $\Omega$, let $\mu$ be the uniform probability measure on $\Omega^n$. Say that two words $w=(w_1,\dots,w_n),u=(u_1,\dots,u_n)\in\Omega^n$ overlap if there exists $j\in\{1,\dots,n\}$ such that the final segment $(w_{n-j+1},\dots,w_n)$ equals the initial segment $(u_1,\dots,u_j)$. For $A\subset\Omega^n$, define
\[
U(A)=\{u\in\Omega^n:\text{for every }w\in A,\ (w,u)\text{ does not overlap}\}.
\]
Now define $\gamma(\alpha,n)$ to be the largest possible value of $\mu(U(A))$ over all finite sets $\Omega$ and all subsets $A\subset\Omega^n$ with $\mu(A)=\alpha$. Which quantitative estimate holds for $\gamma(\alpha,n)$?

For every $n\ge 1$ and every $\alpha\in(0,1)$,
\[
\gamma(\alpha,n)\le \frac{(1-\alpha)\bigl(1-(1-\alpha)^n\bigr)}{\alpha n}.
\]

Choices:
B.
For every $n\ge 1$ and every $\alpha\in(0,1)$,
\[
\gamma(\alpha,n)\le \frac{1-(1-\alpha)^n}{\alpha n}.
\]
C.
For every $n\ge 1$ and every $\alpha\in(0,1)$,
\[
\gamma(\alpha,n)\le \frac{1}{\alpha n}.
\]
D.
For every $n\ge 1$ and every $\alpha\in(0,1)$,
\[
\gamma(\alpha,n)\le \frac{(1-\alpha)^n}{\alpha n}.
\]
E.
For every $n\ge 1$ and every $\alpha\in(0,1)$,
\[
\gamma(\alpha,n)= \frac{(1-\alpha)\bigl(1-(1-\alpha)^n\bigr)}{\alpha n}.
\]

missing prefactor $(1-\alpha)$ from the recursive/random-walk estimate

dropped the numerator refinement $(1-\alpha)(1-(1-\alpha)^n)$ and kept only its trivial upper bound by $1$

replaced the cumulative factor $1-(1-\alpha)^n$ by the terminal term $(1-\alpha)^n$

turned the upper bound from the recursive decomposition into an exact formula for all alphabets and sets


\label{thm:main-intro}
    Assume $\mathcal{A}$ is symmetric and $\beta$-stable with respect to $\mathcal{A}^*$, that $D_{1:n+1}$ is exchangeable, and that
\begin{equation}
        \E\left[\ell(X_{n+1}, Y_{n+1}; \mathcal{A}^*(D_{1:n+1})) \right] \leq \alpha-\beta.
\end{equation}
    Then
    \begin{equation}
        \E\left[\ell(X_{n+1}, Y_{n+1}; \mathcal{A}(D_{1:n})) \right] \leq \alpha.
    \end{equation}

\begin{theorem}
    \label{thm:main-intro}
    Assume $\A$ is symmetric and $\beta$-stable with respect to $\A^*$, that $D_{1:n+1}$ is exchangeable, and that
\begin{equation}
        \E\left[\ell(X_{n+1}, Y_{n+1}; \A^*(D_{1:n+1})) \right] \leq \alpha-\beta.
\end{equation}
    Then
    \begin{equation}
        \E\left[\ell(X_{n+1}, Y_{n+1}; \A(D_{1:n})) \right] \leq \alpha.
    \end{equation}
\end{theorem}

\begin{proposition}
    \label{prop:crc-stable}
    Let $d=1$, $D$ be any dataset, $\ell$ be nonincreasing in its last argument,
    \begin{equation}
        \A(D) = \inf\left\{ \theta : \frac{1}{|D|+1}\sum_{(x,y) \in D}\ell(x,y;\theta) \leq \alpha - \frac{1}{|D|+1} \right\},
    \end{equation}
    and
    \begin{equation}
        \A^*(D) = \inf\left\{ \theta : \frac{1}{|D|}\sum_{(x,y) \in D}\ell(x,y;\theta) \leq \alpha \right\}.
    \end{equation}
    Then $\A$ is $0$-stable with respect to $\A^*$.
\end{proposition}

Consider a sequence of random variables $D_{1:n+1} = ((X_1, Y_1), \ldots, (X_{n+1}, Y_{n+1}))$ representing feature-label pairs with an exchangeable joint distribution.
Let the first $n$ datapoints, $D_{1:n}$, be a known calibration set, and last datapoint, $(X_{n+1},Y_{n+1})$, represent a test datapoint whose label is unknown.
Also define a bounded loss, $\ell(x,y;\theta) \in [0,1]$, which is a function of a datapoint $(x,y)$ and a parameter $\theta \in \R^d$.
Our goal is to select a parameter value $\hat{\theta}$ using the calibration data $D_{1:n}$ to bound the expected loss on the test datapoint:
\begin{equation}
 \label{eq:intro-guarantee}
 \E\left[ \ell(X_{n+1}, Y_{n+1}; \hat{\theta}) \right] \leq \alpha,
\end{equation}
where $\alpha \in [0,1]$ is user-specified and the expectation is taken with respect to all $n+1$ datapoints.
We call~\eqref{eq:intro-guarantee} a risk control guarantee.

Conformal risk control, a generalization of conformal prediction~\cite{gammerman1998learning, vovk1999machine, vovk2005algorithmic,lei2013conformal,lei2018distribution} as developed in~\cite{angelopoulos2024conformal}, handles the case where $d=1$ and $\ell$ is monotonically nonincreasing in $\theta$.
Conformal risk control works by setting the parameter $\hat\theta$ to be
\begin{equation}
 \hat\theta = \inf\left\{ \theta : \frac{1}{n+1}\sum_{i=1}^{n}\ell(X_i,Y_i;\theta) \leq \alpha - \frac{1}{n+1} \right\},
\end{equation}
which is the smallest value of $\theta$ such that the empirical risk calculated on $n+1$ datapoints is certain to be below $\alpha$. This provides the guarantee in~\eqref{eq:intro-guarantee} for monotonic losses, but can fail arbitrarily badly for non-monotonic losses, as shown in Proposition 1 of~\cite{angelopoulos2024conformal}.

In this paper, we give risk control guarantees for non-monotonic losses.
The key insight is that the population risk of $\hat{\theta}$ depends on the stability of the algorithm for choosing $\hat{\theta}$---i.e., the change in risk when the test datapoint is added to or removed from the calibration dataset.
More formally, let $\mathcal{A}$ be an algorithm mapping datasets to choices of $\theta$ and let $D_{-i}$ denote $D_{1:n+1}$ with the $i$th element removed.
The parameter $\hat{\theta}$ is the output of $\mathcal{A}(D_{1:n})$.
We say an algorithm $\mathcal{A}$ is $\beta$-stable with respect to a reference algorithm $\mathcal{A}^*$ and $\ell$ if
\begin{equation}
 \E\left[\frac{1}{n+1}\sum\limits_{i=1}^{n+1} \ell(X_{i}, Y_{i}; \mathcal{A}(D_{-i}))\right]
 \leq \E\left[\frac{1}{n+1}\sum\limits_{i=1}^{n+1} \ell(X_{i}, Y_{i}; \mathcal{A}^*(D_{1:n+1}))\right] + \beta.
\end{equation}
The main theorem demonstrates that if an algorithm is stable with respect to a reference algorithm that controls the risk when run on the full data, then the original algorithm also controls the risk.
Here and throughout the paper, we assume the algorithms are symmetric (i.e. permutation-invariant).

\begin{equation}
    \label{eq:intro-guarantee}
    \E\left[ \ell(X_{n+1}, Y_{n+1}; \thetahat) \right] \leq \alpha,
\end{equation}

\subsection{Conformal Risk Control is Stable for Monotonic Losses}

\subsubsection{General Bounded Losses}
\label{sec:general-bounded}
To handle general losses bounded in $[0,1]$, we will use algorithms that discretize $\Theta$ to impose stability.
First, let $\A(D) = \inf\{\theta \in \Theta_m = \{0, \tfrac{1}{m}, \tfrac{2}{m}, \ldots, 1\} : \hat{R}_D(\theta) \leq \alpha\}$ for $\Theta = [0,1]$ and any positive integer $m$.
This is simply the discretized version of the previously studied algorithms.
We will assume nothing about the loss other than the existence of a safe solution, $\ell(z;1)=0$ for all $z \in \cX\times\cY$.
\begin{proposition}
\label{prop:general-stability}
Let $D_{1:n+1}$ be sampled i.i.d., $\Theta_m=\{0,\tfrac1m,\ldots,1\}$, $\ell(\cdot;1)=0$, and define
\begin{equation}
\A(D)\ :=\ \inf\bigl\{\theta\in\Theta_m:\ \hat R_D(\theta)\le \alpha\bigr\}.
\end{equation}
Then
\begin{equation}
\E\bigl[R(\thetahat_{n})\bigr]\ \leq\ \alpha+\frac{1}{2\sqrt{n}}\left(\sqrt{-W_{-1} \left( -\frac{1}{4n(m+1)^2}\right)}+\sqrt{-\frac{1}{W_{-1} \left( -\frac{1}{4n(m+1)^2}\right)}}\right) = \alpha + \tilde{\mathcal{O}}\left(\frac{1}{\sqrt{n}}\right),
\end{equation}
where $W_{-1}$ is the $-1$st branch of the Lambert W function and $\tilde{\mathcal{O}}$ denotes the growth rate excluding logarithmic factors.
\end{proposition}
This proposition tells us that we can achieve risk control for general losses with the discretized algorithm up to a dominating factor of $\tilde{\mathcal{O}}\left(\tfrac{1}{\sqrt{n}}\right)$, since $W_{-1}$ is of logarithmic order.

\begin{proposition}
 \label{prop:stability-selective}
 The selective classification algorithm in~\eqref{eq:leftmost-root} is $\beta$-stable with
 \begin{equation}
 \beta = \frac{2\max\{\alpha, 1-\alpha\}\E[K]}{n+1}, \text{ where } K = \max_i|\hat{\jmath}_{-i} - \hat{\jmath}_{n+1}|.
 \end{equation}
\end{proposition}
\begin{proof}
 By~\eqref{eq:selective-eq-neq}, it suffices to prove that, almost surely,
 \begin{equation}
 \sum\limits_{i=1}^{n+1} \Big(\ell(X_{i}, Y_{i}; \A(D_{-i})) - \ell(X_{i}, Y_{i}; \A^*(D_{1:n+1}))\Big) = \sum_{i\in [n+1]} w_i\Delta_i
 \leq 2\max\{\alpha, 1-\alpha\}K,
 \end{equation}
 where $\Delta_i = \ind{\A(D_{-i}) \geq \hat P_i} - \ind{\A^*(D_{1:n+1}) \geq \hat P_i}$ and $w_i = \alpha$ for $i \in V^{=}$ and $-(1-\alpha)$ otherwise.
 We can begin by rewriting the $\Delta_i$ terms as
 \begin{align}
 \Delta_i &= \ind{\A(D_{-i}) \geq \hat{P}_i > \A(D_{1:n+1})} - \ind{\A(D_{-i}) < \hat{P}_i \leq \A(D_{1:n+1})} \\
 &= \ind{\hat{\jmath}_{-i} \leq V_i < \hat{\jmath}_{n+1}} - \ind{\hat{\jmath}_{-i} > V_i \geq \hat{\jmath}_{n+1}}.
 \end{align}
 Note that the two events inside the indicators are mutually exclusive, and correspond to the ``crossing'' of $\A(D_{-i})$ over $\hat{P}_i$ in the downward or upward directions.
 Therefore,
 \begin{equation}
 \label{eq:pf-selective-cauchy-schwarz}
 \sum_{i\in [n+1]} w_i\Delta_i \leq \max\{\alpha, 1-\alpha\} \sum_{i\in [n+1]}|\Delta_i| \leq \max\{\alpha, 1-\alpha\}\left|\{i : \hat{\jmath}_{-i} \leq V_i < \hat{\jmath}_{n+1} \text{ or } \hat{\jmath}_{-i} > V_i \geq \hat{\jmath}_{n+1}\} \right|.
 \end{equation}
 Define the index sets
\[
S_- \;:=\; \bigl\{i\in[n{+}1]:\ \hat{\jmath}_{-i}\le V_i<\hat{\jmath}_{n+1}\bigr\}
\quad\text{and}\quad
S_+ \;:=\; \bigl\{i\in[n{+}1]:\ \hat{\jmath}_{-i}> V_i\ge \hat{\jmath}_{n+1}\bigr\}.
\]
Then the set of indices that contribute nonzero terms to $\sum_{i\in[n+1]}|\Delta_i|$ is $S:=S_-\cup S_+$, and $S_-\cap S_+=\varnothing$ by definition.

\subsubsection{Risk Control Guarantees on the Loss}
First, we show the stability of ERM on the scale of the loss.
\begin{proposition}
\label{prop:erm-stable-loss}
Let $\Theta=\R^d$, $\ell$ be convex in $\theta$, and assume there exists $\rho:\cZ\to[0,\infty)$ such that for all $z\in\cZ$ and $\theta,\theta'\in\R^d$,
\begin{equation}
\label{eq:lipschitz-rho-simple}
|\ell(z;\theta)-\ell(z;\theta')|\ \le\ \rho(z)\,\|\theta-\theta'\|_2.
\end{equation}
Then $\A$ is $\beta$-stable with
\begin{equation}
\label{eq:beta-simple}
\beta \leq \frac{2\,\E[\rho(Z_{n+1})^2]}{\lambda(n+1)}.
\end{equation}
\end{proposition}
This result can be used in conjunction with Theorem~\ref{thm:main-intro} to give a generalization guarantee on ERM, as below.
\begin{corollary}
 In the setting of Proposition~\ref{prop:erm-stable-loss},
 \begin{equation}
 \E\left[ \ell(X_{n+1}, Y_{n+1}; \thetahat_n) + \tfrac{\lambda}{2}\|\thetahat_n\|_2^2 \right] \leq R^* + \frac{2\,\E[\rho(Z_{n+1})^2]}{\lambda(n+1)},
 \end{equation}
 where $R^* = \min_{\theta \in \Theta} \E[\ell(X_{n+1},Y_{n+1};\theta)+ \tfrac{\lambda}{2}\|\theta\|_2^2]$.
\end{corollary}
These results represent minor variations on the canonical results in Section 5 of~\cite{bousquet2002stability}.

Next, we address stability of the expected gradient.
Building towards this, we will need a multivariate notion of stability to handle gradients when $d>1$: an algorithm $\cA$ is $\beta$-stable with respect to $\A^*$ and $g$ if
\begin{equation}
 \E\left[\frac{1}{n+1}\sum\limits_{i=1}^{n+1} g(X_{i}, Y_{i}; \A(D_{-i}))\right]
 \preceq \E\left[\frac{1}{n+1}\sum\limits_{i=1}^{n+1} g(X_{i}, Y_{i}; \A^*(D_{1:n+1}))\right] + \beta,
\end{equation}
where $\beta \in \R^d$, and $\preceq$ represents the standard partial ordering on vectors (i.e., the inequality holds component-wise).
Theorem~\ref{thm:main-intro} also has an extension to $d>1$, which we present below.
\begin{theorem}
 \label{thm:main-high-d}
 Assume $\A$ is symmetric and $\beta$-stable with respect to $\A^*$ and $g$, that $D_{1:n+1}$ is exchangeable, and that
\begin{equation}
 \E\left[g(X_{n+1}, Y_{n+1}; \A^*(D_{1:n+1})) \right] \preceq \alpha-\beta.
\end{equation}
 Then
 \begin{equation}
 \E\left[g(X_{n+1}, Y_{n+1}; \A(D_{1:n})) \right] \preceq \alpha.
 \end{equation}
\end{theorem}
The proof of this theorem is identical to that of Theorem~\ref{thm:main-intro}, except with $\preceq$ in place of $\leq$.
Now we proceed with the stability result, stated below for differentiable convex losses. The differentiability is not needed, and is only used to simplify the statement and proof of the proposition.
\begin{proposition}
\label{prop:erm-grad-stability}
Assume $\ell$ is a convex, differentiable function of $\theta$ and that there exists a measurable function $\rho : \cZ \to [0,\infty)$ such that for all $\theta,\theta' \in \R^d$ and almost all $z \in \cZ$,
\begin{equation}
\|\nabla\ell(z;\theta) - \nabla\ell(z;\theta')\|_2 \leq \rho(z)\|\theta - \theta'\|_2.
\end{equation}
Assume also that $\hat{R}_{1:n+1}$ is $\mu$-strongly convex for some $\mu \in [0,\infty)$ (noting that $\mu=0$ is also allowable, so it need not be strongly convex).
Then the regularized ERM algorithm $\cA$ is $\beta$-stable with respect to $\nabla\ell$, where
\begin{equation}
\beta = \frac{\E[\rho(Z_{n+1})\|\nabla\ell(Z_{n+1}; \thetahat_n)\|_2] + \E\left[\rho(Z_{n+1})\|\tfrac{1}{n}\sum_{j=1}^n\nabla\ell(Z_j;\thetahat_n)\|_2\right]}{(\mu + \lambda)(n+1)} \mathbf{1}_d.
\end{equation}
\end{proposition}
Eliding the Lipschitz constant $\rho(Z_{n+1})$, this result tells us that ERM is stable with
\begin{equation}
 \beta = \frac{\E[\text{magnitude of test gradient}] + \E[\text{magnitude of training gradient}]}{(\mu + \lambda)(n+1)}.
\end{equation}
Under normal circumstances, both terms in the numerator will be constant-order, and the latter term will be near-zero (in fact, when $\lambda=0$, it is exactly zero).
Proposition~\ref{prop:erm-grad-stability} also holds for nondifferentiable losses, replacing every gradient in $\beta$ with a supremum over all subgradients in the subdifferential at $\thetahat_{n}$; the proof of this more general result is morally the same, so we omit it.
The assumption that $\hat{R}_{1:n+1}$ is strongly convex is weaker than strong convexity of the loss function itself, and allows for, e.g., ordinary least squares with a full-rank design matrix, even though the individual losses may not be strongly convex.

Theorem~\ref{thm:main-intro} gives us an actionable workflow for producing risk-control algorithms.
First, we try to identify a reference algorithm, $\mathcal{A}^*$, that controls the risk when applied on the full dataset of $n+1$ datapoints.
Then, we try to approximate that algorithm via $\mathcal{A}$, which runs only on $n$ datapoints, and prove that the difference in risks between these two algorithms is low.
The proof of the main result is below.
\begin{proof}
By exchangeability and symmetry,
\[
\mathbb{E}\!\left[\frac{1}{n+1}\sum_{i=1}^{n+1}\ell(X_i,Y_i;\mathcal{A}^*(D_{1:n+1}))\right]
= \mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}^*(D_{1:n+1}))\right],
\]
and
\[
\mathbb{E}\!\left[\frac{1}{n+1}\sum_{i=1}^{n+1}\ell(X_i,Y_i;\mathcal{A}(D_{-i}))\right]
= \mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}(D_{1:n}))\right].
\]
By $\beta$-stability, $\mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}(D_{1:n}))\right] \leq \mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}^*(D_{1:n+1}))\right] + \beta$.
Combining this with the assumption that $\mathbb{E}[\ell(X_{n+1},Y_{n+1};\mathcal{A}^*(D_{1:n+1}))] \leq \alpha-\beta$ yields the result.
\end{proof}
Notice that Theorem~\ref{thm:main-intro} applies to $\theta$ in any space, and that boundedness of $\ell$ was never needed in the proof of Theorem~\ref{thm:main-intro}.
These concerns will come up later in determining stability bounds, building on the literature on algorithmic stability~\cite{kearns1997algorithmic, bousquet2002stability, kutin2002almost, mukherjee2006learning, shalev-shwartz2010learnability, yu2013stability, hardt2016train, yu2017three, feldman2018generalization, bousquet2020sharper, zrnic2023post}; see Chapter 13 of~\cite{shalev-shwartz2014understanding} for a review of this extensive field.
The notion of stability used herein is a form of leave-one-out stability with respect to generic algorithms $\cA$ and $\cA^*$; we will prove guarantees for a few different choices of algorithms depending on the setting.

\subsection{Conformal Risk Control is Stable for Monotonic Losses}

Here we will show that the conformal risk control algorithm is stable for monotonic losses, and thus, satisfies the guarantee in Theorem~\ref{thm:main-intro}.

\begin{proposition}
    \label{prop:crc-stable}
    Let $d=1$, $D$ be any dataset, $\ell$ be nonincreasing in its last argument,
    \begin{equation}
        \mathcal{A}(D) = \inf\left\{ \theta : \frac{1}{|D|+1}\sum_{(x,y) \in D}\ell(x,y;\theta) \leq \alpha - \frac{1}{|D|+1} \right\},
    \end{equation}
    and
    \begin{equation}
        \mathcal{A}^*(D) = \inf\left\{ \theta : \frac{1}{|D|}\sum_{(x,y) \in D}\ell(x,y;\theta) \leq \alpha \right\}.
    \end{equation}
    Then $\mathcal{A}$ is $0$-stable with respect to $\mathcal{A}^*$.
\end{proposition}
\begin{proof}
    The boundedness of $\ell$ and definition of $\mathcal{A}$ imply that for all $i$,
    \begin{equation}
        \frac{1}{n+1}\sum_{j=1}^{n+1}\ell(X_j,Y_j;\mathcal{A}(D_{-i})) \leq \frac{1}{n+1}\sum_{j \in [n+1] \setminus i} \ell(X_j,Y_j;\mathcal{A}(D_{-i})) + \frac{1}{n+1} \leq \alpha.
    \end{equation}
    $\mathcal{A}(D_{-i})$ therefore satisfies the constraint in the definition of $\mathcal{A}^*(D_{1:n+1})$, and the monotonicity of the loss gives $\mathcal{A}(D_{-i}) \geq \mathcal{A}^*(D_{1:n+1})$. Therefore, $\ell(X_{i}, Y_{i}; \mathcal{A}(D_{-i})) \leq \ell(X_{i}, Y_{i}; \mathcal{A}^*(D_{1:n+1}))$, and
    \begin{equation}
        \E\left[\frac{1}{n+1}\sum\limits_{i=1}^{n+1} \Big(\ell(X_{i}, Y_{i}; \mathcal{A}(D_{-i})) - \ell(X_{i}, Y_{i}; \mathcal{A}^*(D_{1:n+1}))\Big)\right] \leq 0.
    \end{equation}
    Rearranging terms gives $0$-stability.
\end{proof}

Combining Proposition~\ref{prop:crc-stable} with Theorem~\ref{thm:main-intro} exactly recovers the classical conformal risk control guarantee.
\begin{corollary}
    The conformal risk control algorithm $\mathcal{A}$ defined in Proposition~\ref{prop:crc-stable} satisfies
    \begin{equation}
        \E\left[ \ell(X_{n+1},Y_{n+1};\mathcal{A}(D_{1:n})) \right] \leq \alpha.
    \end{equation}
\end{corollary}
\begin{proof}
    For monotonic losses, Proposition~\ref{prop:crc-stable} shows that the conformal risk control algorithm $\mathcal{A}$ is $0$-stable. Applying Theorem~\ref{thm:main-intro} with $\beta=0$ recovers the guarantee.
\end{proof}

The key takeaway is that the proof of conformal risk control's validity has two parts, which can be decoupled: (1) proving monotonicity implies stability, and (2) proving that stability implies risk control.
Theorem~\ref{thm:main-intro} tells us that (2) holds for general algorithms.
The remainder of the paper will be dedicated to replacing step (1) with other stable algorithms, in the context of non-monotonic losses.
Looking forward, Section~\ref{sec:methods} gives stability bounds for classes of non-monotonic losses and shows how to estimate these stability bounds for use in Theorem~\ref{thm:main-intro}. Section~\ref{sec:experiments} shows experimental validation of these methods.

To prove Theorem~\ref{thm:main-intro}, the text argues:

1. **Use exchangeability and symmetry to rewrite the target expectations as averages.** By exchangeability and symmetry,
\[
\mathbb{E}\!\left[\frac{1}{n+1}\sum_{i=1}^{n+1}\ell(X_i,Y_i;\mathcal{A}^*(D_{1:n+1}))\right]
= \mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}^*(D_{1:n+1}))\right],
\]
and similarly
\[
\mathbb{E}\!\left[\frac{1}{n+1}\sum_{i=1}^{n+1}\ell(X_i,Y_i;\mathcal{A}(D_{-i}))\right]
= \mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}(D_{1:n}))\right].
\]

2. **Apply $\beta$-stability to compare the risks of $\mathcal{A}$ and $\mathcal{A}^*$.** The proof states: by $\beta$-stability,
\[
\mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}(D_{1:n}))\right] \leq \mathbb{E}\!\left[\ell(X_{n+1},Y_{n+1};\mathcal{A}^*(D_{1:n+1}))\right] + \beta.
\]

3. **Use the assumption on $\mathcal{A}^*$ to conclude.** Combining the above with the assumption
\(\mathbb{E}[\ell(X_{n+1},Y_{n+1};\mathcal{A}^*(D_{1:n+1}))] \leq \alpha-\beta\)
“yields the result,” i.e.,
\(\mathbb{E}[\ell(X_{n+1},Y_{n+1};\mathcal{A}(D_{1:n}))] \leq \alpha\).

The post-theorem discussion also notes that Theorem~\ref{thm:main-intro} “applies to $\theta$ in any space,” and that “boundedness of $\ell$ was never needed in the proof of Theorem~\ref{thm:main-intro}.”

Question:
Let $D_{1:n+1}=((X_1,Y_1),\ldots,(X_{n+1},Y_{n+1}))$ be feature-label pairs with an exchangeable joint distribution, and let $\ell(x,y;\theta)$ be a loss function. For each $i$, let $D_{-i}$ denote $D_{1:n+1}$ with its $i$th datapoint removed. Suppose $\mathcal A$ and $\mathcal A^*$ are symmetric (permutation-invariant) algorithms mapping datasets to parameter values, and suppose $\mathcal A$ is $\beta$-stable with respect to $\mathcal A^*$ and $\ell$ in the sense that
\[
\mathbb E\!\left[\frac{1}{n+1}\sum_{i=1}^{n+1} \ell(X_i,Y_i;\mathcal A(D_{-i}))\right]
\le
\mathbb E\!\left[\frac{1}{n+1}\sum_{i=1}^{n+1} \ell(X_i,Y_i;\mathcal A^*(D_{1:n+1}))\right] + \beta.
\]
Assume also that
\[
\mathbb E\bigl[\ell(X_{n+1},Y_{n+1};\mathcal A^*(D_{1:n+1}))\bigr] \le \alpha-\beta.
\]
Under these assumptions, which quantitative estimate holds for the test-point loss of $\mathcal A$ trained on $D_{1:n}$?

\[
\mathbb E\bigl[\ell(X_{n+1},Y_{n+1};\mathcal A(D_{1:n}))\bigr] \le \alpha.
\]

Choices:
B.
\[
\mathbb E\bigl[\ell(X_{n+1},Y_{n+1};\mathcal A(D_{1:n}))\bigr] \le \alpha-\beta.
\]
C.
\[
\mathbb E\bigl[\ell(X_{n+1},Y_{n+1};\mathcal A(D_{1:n}))\bigr] \le \alpha+\beta.
\]
D.
\[
\mathbb E\bigl[\ell(X_{n+1},Y_{n+1};\mathcal A(D_{1:n+1}))\bigr] \le \alpha.
\]
E.
\[
\mathbb E\!\left[\frac{1}{n+1}\sum_{i=1}^{n+1} \ell(X_i,Y_i;\mathcal A(D_{-i}))\right] \le \alpha.
\]

dropped the sharp conclusion threshold from \(\alpha\) to the weaker bound \(\alpha+\beta\)

leave-one-out average substituted for test-point conclusion


\begin{thm}\label{thm:mainisoperim}
For all $A\subset \{0,1\}^n$ with $|A|\le \frac12$,
\begin{equation}\label{eqn:isoperimhalf}
\mathbf{E} \sqrt{h_A} \ge |A| \sqrt{\log_2(1/|A|)}.
\end{equation}
\end{thm}

Let $n\ge 1$ be an integer.
For $A\subset \{0,1\}^n$ and $x\in A$ let
$h_A(x)$ denote the number of edges connecting $x$ with $\{0,1\}^n\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.
If $x\not\in A$, then let $h_A(x)=0$.
Here $\mathbf{E}f=2^{-n} \sum_{x\in \{0,1\}^n} f(x)$ and $|A|=\mathbf{E} \mathbf{1}_A$.

\section{Introduction}
Let $n\ge 1$ be an integer.
For $A\subset \{0,1\}^n$ and $x\in A$ let
$h_A(x)$ denote the number of edges connecting $x$ with $\{0,1\}^n\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.
If $x\not\in A$, then let $h_A(x)=0$.
Here $\mathbf{E}f=2^{-n} \sum_{x\in \{0,1\}^n} f(x)$ and $|A|=\mathbf{E} \mathbf{1}_A$.

This inequality is sharp in two different ways: first, it is an equality when $A$ is a subcube.
Second, there is no corresponding dimension-free lower bound when $\sqrt{h_A}$ is replaced by $h_A^\beta$ for any $\beta<\frac12$, which can be seen by Hamming ball examples.
The study of $\mathbf{E} \sqrt{h_A}$ was first initiated by Talagrand \cite{Tal93} who proved lower bounds such as $\mathbf{E}\sqrt{h_A}\ge \sqrt{2}|A|(1-|A|)$. This was later improved by Bobkov--G\"otze \cite{BG99}, as a consequence of more general results.

Sharp lower bounds for $\mathbf{E} h_A^\beta$ for all $\beta\ge \frac12$ follow from \eqref{eqn:isoperimhalf} and H\"older's inequality; see e.g. \cite[Lemma 5.1]{DIR24}.
Such bounds were previously proved for $\beta\ge 0.50057$ in \cite{DIR24} and prior to that for $\beta\ge \log_2(3/2)\approx 0.585$ in \cite{KP20,BIM23}.
For $\beta=1$ one has Harper's classical isoperimetric inequality \cite{Har66,Ber67,Hart76}.
By the Cauchy-Schwarz inequality, \eqref{eqn:isoperimhalf} implies the following sharp strengthening of the classical isoperimetric inequality for $|A|\le \frac12$,
\begin{equation}\label{eqn:classicalsharpening}
\mathbf{E} h_A \ge \frac{|A|}{|\partial A|} |A|\log_2(1/|A|),
\end{equation}
where $\partial A$ is the support of $h_A$, i.e. the vertex boundary of $A$.
This is an equality if $A$ is a subcube. It improves on the classical isoperimetric inequality because of the factor $\frac{|A|}{|\partial A|}\ge 1$. Moreover, it fails when the factor is replaced by $(\frac{|A|}{|\partial A|})^{\gamma}$ for any $\gamma>1$, again by Hamming ball examples.
Such an inequality was observed by Beltran--Ivanisvili--Madrid \cite[Cor. 1.2]{BIM23} for $\gamma=\frac1{\log_2 (3/2)}-1\approx 0.709$.

\subsection*{Applications}
As an immediate consequence of Theorem \ref{thm:mainisoperim} we settle a conjecture of Kahn and Park \cite[Conjecture 1.3]{KP20} on cube partitions:
\begin{thm}\label{cor:kahnpark}
Let $(A,B,W)$ be a partition of $\{0,1\}^n$ and assume $|A|=\frac12$. Then
\[ |\nabla(A,B)| + \sqrt{n}\,|W|\ge \tfrac12. \]
\end{thm}
Here $|W|=\mathbf{E} \mathbf{1}_W$ and the edge boundary measure $|\nabla(A,B)|$ is defined by $2^{-n} \# \{(x,y)\in \mathcal{E}\,:\,x\in A, y\in B\}$, where $\mathcal{E}$ is the set of edges in the Hamming cube $\{0,1\}^n$.
Theorem \ref{cor:kahnpark} follows from Theorem \ref{thm:mainisoperim} because for $|B\cup W|=|A|=\frac12$, we have
\[\tfrac12\le \mathbf{E} \sqrt{h_{B\cup W}} \le \mathbf{E} (h_{B\cup W}\mathbf{1}_B) + \mathbf{E} (\sqrt{h_{B\cup W}}\mathbf{1}_W),\]
which is no greater than $|\nabla(A,B)| + \sqrt{n} |W|$ as required.

Our next application is a sharp $L^1$ Poincar\'e inequality for Boolean-valued functions.
\begin{thm}\label{thm:poincare} For all $f:\{0,1\}^n\to \{0,1\}$,
\begin{equation}\label{eqn:poincareL1}
\|\nabla f\|_1\ge \|f-\mathbf{E}f\|_1.
\end{equation}
Equality is achieved when $f=\mathbf{1}_A$ for a half-cube $A$.
\end{thm}
Here $|\nabla f(x)|=\sqrt{\sum_{i=1}^n (\tfrac12(f(x)-f(x\oplus e_i)))^2}$ and $\|f\|_1=\mathbf{E} |f|$. Note that it is still open to determine the sharp constant in this inequality for {\it real-valued} $f$, but it is known that this constant lies in the interval $(\frac2{\pi},\sqrt{\frac{2}{\pi}}]$; see \cite{BELP08}, \cite{ILvHV}, also \cite{haonan1}, \cite{Esken1} for quantum and vector-valued inequalities. In particular, \eqref{eqn:poincareL1} shows that the lower bound improves for Boolean-valued functions, which was first observed in \cite{BIM23}.
Theorem \ref{thm:poincare} does not follow directly from Theorem \ref{thm:mainisoperim}, but from the more technical isoperimetric inequality \eqref{eqn:isoperim2} proved below.

Finally, isoperimetric inequalities for the Hamming cube are also related to a family of conjectures in information theory concerning noisy channels (Chen--Nair \cite{CN24}). One of these is the most informative Boolean function conjecture, also known as Courtade--Kumar conjecture \cite{KC13}.
The latter would be implied by a stronger conjecture, known as Hellinger conjecture \cite{ABCJN17, CGN25}. Roughly speaking, these conjectures concern optimality questions for Boolean functions under standard noisy-channel models on $\{0,1\}^n$, where each bit is independently perturbed.
In the low-noise limit, the Hellinger conjecture would in particular imply that
\begin{equation}\label{eqn:sensitivity}
\mathbf{E}\sqrt{s_f} \ge 1
\end{equation}
holds for balanced Boolean functions $f:\{0,1\}^n\to \{0,1\}$ (i.e. $\mathbf{E}f=\tfrac12$), where $s_f(x)$ is the sensitivity of $f$ at $x$, which is defined as the number of single-bit flips of $x$ that change the value of $f(x)$, i.e. $s_f(x)=h_A(x)+h_{A^c}(x)$.
This implication is well-known, but for convenience of the reader we provide details in \S \ref{sec:it}.
Since $h_A$ and $h_{A^c}$ have disjoint support, $\sqrt{s_f}=\sqrt{h_A}+\sqrt{h_{A^c}}$, so Theorem \ref{thm:mainisoperim} with $|A|=\frac12$ immediately implies \eqref{eqn:sensitivity}.

For $0\le x\le y\le 1$ and $B:[0,1]\to [0,\infty)$ let
\begin{equation}\label{eqn:Gdef}
G_1[B](x,y) = \sqrt{(y-x)^{2}+B(y)^{2}} + B(x) - 2 B(\tfrac{x+y}2),
\end{equation}
\[ G_2[B](x,y) = y-x + (\sqrt{2}-1) B(y) + B(x) - 2 B(\tfrac{x+y}2),\]
and $G = \max(G_1, G_2).$
Kahn and Park \cite{KP20} proved that if $B$ satisfies $B(0)=B(1)=0$ and the two-point inequality
\[G[B](x,y)\ge 0\]
holds for all $0\le x\le y\le 1$, then $\mathbf{E} \sqrt{h_A}\ge B(|A|)$
for all $A\subset \{0,1\}^n$ and $n\ge 1$.

This reduces the proof of Theorem \ref{thm:mainisoperim} to finding a Bellman function $B$ and verifying the two-point inequality.
We use the following refinement of the Bellman function used in \cite{DIR24}.
For $x\in (0,1]$ define
\[ L(x) = x \sqrt{\log_2(1/x)} \]
and $L(0)=0$.
Let $Q(x)$ be the unique cubic interpolating polynomial such that $Q(0)=Q(1)=0$, $Q(\frac12)=\frac12$ and $Q(\frac14)=2^{-3/2}$, i.e.
\[ Q(x) = \tfrac23 x (1-x)(2^{5/2}-3+4(3-2^{3/2}) x). \]
The idea of using the polynomial $Q$ goes back to Beltran--Ivanisvili--Madrid \cite{BIM23}.
Let $I(x)$ denote the Gaussian isoperimetric profile, i.e. the unique function on $[0,1]$ such that
$I(0)=I(1)=0, I\cdot I''=-1$.
For a parameter $w\in (\frac12,1]$ let
\[ {\mathrm{J}}_w(x) = \tfrac12 I(\tfrac1{2w})^{-1} I(\tfrac{1-x}{w}). \]
Note that ${\mathrm{J}}_w(\frac12)=\frac12$ and ${\mathrm{J}}_w(1)=0$.
Then define
\begin{equation}\label{eqn:bbdef}
B_w(x) = \left\{ \begin{array}{ll}
L(x) & \text{for } x\in [0, \frac14],\\
Q(x)
& \text{for }x\in [\frac14, \frac12],\\
{\mathrm{J}}_w(x) & \text{for }x\in [\frac12, 1].
\end{array}\right.
\end{equation}
For some motivation on the choice of such functions, see \cite[\S 2]{DIR24}.
In the following we shall fix
\begin{equation}\label{eqn:w1def}
w=w_1=\tfrac{29}{32}
\end{equation}
and write ${\mathrm{J}}={\mathrm{J}}_{w_1}$.
Note that since $B_w(x)\ge L(x)$ for $x\in [0,\frac12]$ (see \cite[Lemma 5.9 (2)]{DIR24}), in order to prove Theorem \ref{thm:mainisoperim} it suffices to prove that there exists $w$ such that
\begin{equation}\label{eqn:maintwopt}
G[B_w](x,y)\ge 0
\end{equation}
for all $0\le x\le y\le 1$.
As a consequence of this argument we obtain the isoperimetric inequality
\begin{equation}\label{eqn:isoperim2}
\mathbf{E} \sqrt{h_A} \ge B_{w_1}(|A|),
\end{equation}
valid for all $A\subset \{0,1\}^n$.

This inequality is sharp in two different ways: first, it is an equality when $A$ is a subcube.
Second, there is no corresponding dimension-free lower bound when $\sqrt{h_A}$ is replaced by $h_A^\beta$ for any $\beta<\frac12$, which can be seen by Hamming ball examples.
The study of $\mathbf{E} \sqrt{h_A}$ was first initiated by Talagrand \cite{Tal93} who proved lower bounds such as $\mathbf{E}\sqrt{h_A}\ge \sqrt{2}|A|(1-|A|)$. This was later improved by Bobkov--G\"otze \cite{BG99}, as a consequence of more general results.

Sharp lower bounds for $\mathbf{E} h_A^\beta$ for all $\beta\ge \frac12$ follow from \eqref{eqn:isoperimhalf} and H\"older's inequality; see e.g. \cite[Lemma 5.1]{DIR24}.
Such bounds were previously proved for $\beta\ge 0.50057$ in \cite{DIR24} and prior to that for $\beta\ge \log_2(3/2)\approx 0.585$ in \cite{KP20,BIM23}.
For $\beta=1$ one has Harper's classical isoperimetric inequality \cite{Har66,Ber67,Hart76}.
By the Cauchy-Schwarz inequality, \eqref{eqn:isoperimhalf} implies the following sharp strengthening of the classical isoperimetric inequality for $|A|\le \frac12$,
\begin{equation}\label{eqn:classicalsharpening}
\mathbf{E} h_A \ge \frac{|A|}{|\partial A|} |A|\log_2(1/|A|),
\end{equation}
where $\partial A$ is the support of $h_A$, i.e. the vertex boundary of $A$.
This is an equality if $A$ is a subcube. It improves on the classical isoperimetric inequality because of the factor $\frac{|A|}{|\partial A|}\ge 1$. Moreover, it fails when the factor is replaced by $(\frac{|A|}{|\partial A|})^{\gamma}$ for any $\gamma>1$, again by Hamming ball examples.
Such an inequality was observed by Beltran--Ivanisvili--Madrid \cite[Cor. 1.2]{BIM23} for $\gamma=\frac1{\log_2 (3/2)}-1\approx 0.709$.

The proof of Theorem \ref{thm:mainisoperim} relies on a crucial induction argument of Kahn and Park \cite{KP20}, which reduced proving lower bounds for $\mathbf{E} h_A^\beta$ to finding a certain Bellman function and verifying a two-point inequality.
Our Bellman function is a further refinement of the function used in \cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \cite{BIM23}.
In the proof of the two-point inequality we draw from estimates and tools introduced in \cite{DIR24}.

\subsection*{Applications}
As an immediate consequence of Theorem \ref{thm:mainisoperim} we settle a conjecture of Kahn and Park \cite[Conjecture 1.3]{KP20} on cube partitions:
\begin{thm}\label{cor:kahnpark}
Let $(A,B,W)$ be a partition of $\{0,1\}^n$ and assume $|A|=\frac12$. Then
\[ |\nabla(A,B)| + \sqrt{n}\,|W|\ge \tfrac12. \]
\end{thm}
Here $|W|=\mathbf{E} \mathbf{1}_W$ and the edge boundary measure $|\nabla(A,B)|$ is defined by $2^{-n} \# \{(x,y)\in \mathcal{E}\,:\,x\in A, y\in B\}$, where $\mathcal{E}$ is the set of edges in the Hamming cube $\{0,1\}^n$.
Theorem \ref{cor:kahnpark} follows from Theorem \ref{thm:mainisoperim} because for $|B\cup W|=|A|=\frac12$, we have
\[\tfrac12\le \mathbf{E} \sqrt{h_{B\cup W}} \le \mathbf{E} (h_{B\cup W}\mathbf{1}_B) + \mathbf{E} (\sqrt{h_{B\cup W}}\mathbf{1}_W),\]
which is no greater than $|\nabla(A,B)| + \sqrt{n} |W|$ as required.

Our next application is a sharp $L^1$ Poincar\'e inequality for Boolean-valued functions.
\begin{thm}\label{thm:poincare} For all $f:\{0,1\}^n\to \{0,1\}$,
\begin{equation}\label{eqn:poincareL1}
\|\nabla f\|_1\ge \|f-\mathbf{E}f\|_1.
\end{equation}
Equality is achieved when $f=\mathbf{1}_A$ for a half-cube $A$.
\end{thm}
Here $|\nabla f(x)|=\sqrt{\sum_{i=1}^n (\tfrac12(f(x)-f(x\oplus e_i)))^2}$ and $\|f\|_1=\mathbf{E} |f|$. Note that it is still open to determine the sharp constant in this inequality for {\it real-valued} $f$, but it is known that this constant lies in the interval $(\frac2{\pi},\sqrt{\frac{2}{\pi}}]$; see \cite{BELP08}, \cite{ILvHV}, also \cite{haonan1}, \cite{Esken1} for quantum and vector-valued inequalities. In particular, \eqref{eqn:poincareL1} shows that the lower bound improves for Boolean-valued functions, which was first observed in \cite{BIM23}.
Theorem \ref{thm:poincare} does not follow directly from Theorem \ref{thm:mainisoperim}, but from the more technical isoperimetric inequality \eqref{eqn:isoperim2} proved below.

Finally, isoperimetric inequalities for the Hamming cube are also related to a family of conjectures in information theory concerning noisy channels (Chen--Nair \cite{CN24}). One of these is the most informative Boolean function conjecture, also known as Courtade--Kumar conjecture \cite{KC13}.
The latter would be implied by a stronger conjecture, known as Hellinger conjecture \cite{ABCJN17, CGN25}. Roughly speaking, these conjectures concern optimality questions for Boolean functions under standard noisy-channel models on $\{0,1\}^n$, where each bit is independently perturbed.
In the low-noise limit, the Hellinger conjecture would in particular imply that
\begin{equation}\label{eqn:sensitivity}
\mathbf{E}\sqrt{s_f} \ge 1
\end{equation}
holds for balanced Boolean functions $f:\{0,1\}^n\to \{0,1\}$ (i.e. $\mathbf{E}f=\tfrac12$), where $s_f(x)$ is the sensitivity of $f$ at $x$, which is defined as the number of single-bit flips of $x$ that change the value of $f(x)$, i.e. $s_f(x)=h_A(x)+h_{A^c}(x)$.
This implication is well-known, but for convenience of the reader we provide details in \S \ref{sec:it}.
Since $h_A$ and $h_{A^c}$ have disjoint support, $\sqrt{s_f}=\sqrt{h_A}+\sqrt{h_{A^c}}$, so Theorem \ref{thm:mainisoperim} with $|A|=\frac12$ immediately implies \eqref{eqn:sensitivity}.

\subsection*{Acknowledgments} P.D., P.I. and J.R. thank the American Institute of Mathematics (AIM) for funding our SQuaRE project. The authors were also supported in part by grants from the National Science Foundation CAREER-DMS-2152401 (P.I.), DMS-2154835 (J.R.) and by the Simons Foundation
SFI-MPS-TSM-00013943 (P.D.), SFI-MPS-TSM-00014075 (J.R.), Simons Fellowship, and Humboldt Research Fellowship (P.I.).

The post-theorem introduction states that the proof of Theorem~\ref{thm:mainisoperim} "relies on a crucial induction argument of Kahn and Park \cite{KP20}, which reduced proving lower bounds for $\mathbf{E} h_A^\beta$ to finding a certain Bellman function and verifying a two-point inequality." It further says: "Our Bellman function is a further refinement of the function used in \cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \cite{BIM23}." Finally, it notes that "[i]n the proof of the two-point inequality we draw from estimates and tools introduced in \cite{DIR24}."

The post-theorem introduction states that the proof of Theorem~\ref{thm:mainisoperim} “relies on a crucial induction argument of Kahn and Park \cite{KP20}, which reduced proving lower bounds for $\mathbf{E} h_A^\beta$ to finding a certain Bellman function and verifying a two-point inequality.” It further says: “Our Bellman function is a further refinement of the function used in \cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \cite{BIM23}.” Finally, it notes that “[i]n the proof of the two-point inequality we draw from estimates and tools introduced in \cite{DIR24}.”

Question: Let \(n\ge 1\) be an integer, and let \(A\subset \{0,1\}^n\). For \(x\in A\), define \(h_A(x)\) to be the number of neighbors of \(x\) in the Hamming cube that lie outside \(A\), equivalently the number of single-bit flips of \(x\) that leave \(A\); for \(x\notin A\), set \(h_A(x)=0\). For any function \(f:\{0,1\}^n\to\mathbb{R}\), write \(\mathbf{E}f=2^{-n}\sum_{x\in\{0,1\}^n} f(x)\), and let \(|A|=\mathbf{E}\mathbf{1}_A=2^{-n}\#A\) denote the normalized size of \(A\). If \(|A|\le \tfrac12\), which quantitative estimate holds?

For every \(A\subset \{0,1\}^n\) with \(|A|\le \tfrac12\), one has \[\mathbf{E}\sqrt{h_A}\ge |A|\sqrt{\log_2(1/|A|)}.\]

Choices:
B. For every \(A\subset \{0,1\}^n\) with \(|A|\le \tfrac12\), one has \[\mathbf{E}\sqrt{h_A}\ge |A|\sqrt{\log_2\!igl(1/(1-|A|)\bigr)}.\]
C. For every \(A\subset \{0,1\}^n\) with \(|A|\le \tfrac12\), one has \[\mathbf{E}\sqrt{h_A}\ge |A|.\]
D. There exists an absolute constant \(c>0\) such that for every \(n\ge 1\) and every \(A\subset \{0,1\}^n\) with \(|A|\le \tfrac12\), one has \[\mathbf{E}\sqrt{h_A}\ge c\,|A|\sqrt{\log_2(1/|A|)}.\]
E. For every \(A\subset \{0,1\}^n\) with \(|A|\le \tfrac12\), one has \[\mathbf{E} h_A\ge |A|\log_2(1/|A|).\]

Bellman-profile dependence on |A| via log_2(1/|A|)

sharp constant 1 replaced by unspecified absolute constant

confuses the sqrt-bound with a first-moment bound of the same sharp form