\label{T_main}
    Let $A\subset\mathbb{R}^d$ ($d\geq 3$) be a set with positive reach with $\dim A\leq 2$ and let $a\in A$ be such that $\dim(\operatorname{Tan}(A,a))=1$. Then there exist a $\operatorname{B}$-set $B\subset\mathbb{R}^2\subset\mathbb{R}^d$, a multirotation $\rho:\mathbb{R}^d\to\mathbb{R}^d$ associated with $T_1(B)$ and a $C^{1,1}$-diffeomorphism $\Phi:U\subset\mathbb{R}^d\to\mathbb{R}^d$ such that $0\in U$, $a=\Phi(0)$, $\rho(B)\subset U$ and $\Phi(\rho(B))$ is a neighbourhood of $a$ in $A$.

    Moreover, if $\Phi$, $B$ and $\rho$ are as above, then $\Phi(\rho(B))$ agrees on a neighbourhood of $a$ with some set $A\subset\mathbb{R}^d$ with positive reach satisfying $\dim A\leq 2$ and $\dim(\operatorname{Tan}(A,a))=1$.

\begin{lemma}\label{local}
For $A \subset \R^d$ and $a \in A$, the following are equivalent.
\begin{enumerate}
\item[(i)]  $  \reach(A,a) > 0.$
\item[(ii)]  There exists $\delta>0$ such that  $A \cap \overline{B}(a,\delta)$ has positive reach.
\item [(iii)]
There exists a  set  $C$ and $\omega>0$ such that  $\reach C>0$ and $A \cap B(a, \omega)= C \cap B(a, \omega)$.
\end{enumerate}
\end{lemma}

\begin{definition}\label{plochy}
	 We say that $A \subset \R^d$ is a {\it  $k$-dimensional $C^{1,1}$
	 surface} ($0<k<d$), if there exist a $k$-dimensional subspace $W$ of $\R^d$ and a
	   $C^{1,1}$  mapping $\vf: W \to W^{\perp}=:V$ such that
		  $A = \{w+\vf(w): \ w \in W\}$.
		Then we will also say that $A$ is a   $C^{1,1}$  surface {\it associated with $V$}.
		\end{definition}

\begin{theorem}\label{T1+}
	Let $d\geq 2$,  $A \subset \R^d$ have positive reach and $a\in T_1^+(A)$. Then there exists
	a  $1$-dimensional $C^{1,1}$ surface
	 $\Gamma$ and
	$\delta >0$
	 such that
	\begin{equation}\label{intj}
	 T_1(A)\cap B(a, \delta) \subset \Gamma \cap B(a, \delta) \subset  A.
	\end{equation}
\end{theorem}

\begin{corollary}  \label{count}
    Let $A\subset\R^d$ be a set with positive reach and $1\leq k< d$. Then
    \begin{enumerate}
        \item[(i)] $T_k(A)$ can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces,
        \item[(ii)] if $\dim A= k$ then $A$ can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces.
    \end{enumerate}
\end{corollary}

\begin{lemma}\label{lipna2}
	Let  $A\subset \R^d$, $0<r< \reach A$,
	$\diam A < r$ and
	 $A \subset T_1^-(A)  \cup   T_2(A)$.
	 Then  $\psi_2:=\psi_2^A$  is $(2^{12}\pi/r)$-Lipschitz on $T_2(A)$.
\end{lemma}

\begin{equation}  \label{reach_compact}
    \reach(K,a)>0\text{ for all }a\in K\iff\reach K>0.
\end{equation}

\begin{theorem}\label{tdmnapl2}
	Let $d\geq 2$, $1\leq k \leq d-1$,  $A \subset \R^d$ have positive reach and $a\in T_k(A)$.
	Then there exists $\delta >0$
	 such that  $T_k(A) \cap B(a, \delta)$ is a subset of a   $k$-dimensional $C^{1,1}$ surface
	 $\Gamma$.
\end{theorem}

\begin{definition}[B-sets]  \label{B-sets}
\begin{enumerate}
    \item[(i)] We call $M\subset \R^2$  a \emph{$\B_-$-set} if there exist $r>0$, Lipschitz functions $\psi \leq 0 \leq \vf$ on $[0,r]$ such that $\vf$ is semiconcave on $(0,r)$, $\psi$ is semiconvex on $(0,r)$, $\vf(0)= \psi(0)=0$, $\vf'_+(0)= \psi'_+(0)=0$ and $M= \{(x,y):\ x\in [0,r],  \psi(x) \leq y \leq \vf(x)\}$.
    \item[(ii)] We call $M\subset \R^2$  a \emph{$\B_+$-set} if there exist $r>0$, Lipschitz functions $\psi \leq 0 \leq \vf$ on $[-r,r]$ such that $\vf$ is semiconcave on $(-r,r)$, $\psi$ is semiconvex on $(-r,r)$, $\vf(0)= \psi(0)=0$, $\vf'(0)= \psi'(0)=0$ and $M= \{(x,y):\ x\in [-r,r],  \psi(x) \leq y \leq \vf(x)\}$.
    \item[(iii)] We say that $M\subset\R^2$ is a \emph{$\B$-set} if $M$ is a $\B_-$-set or a $\B_+$-set.
\end{enumerate}
\end{definition}

A set $A\subset\mathbb{R}^d$ is said to have \emph{positive reach} if for some $\varepsilon>0$, any point $x\in\mathbb{R}^d$ with $\mathrm{dist}(x,A)<\varepsilon$ have its unique nearest point $a\in A$ (point with $\mathrm{dist}(x,A)=|x-a|$). Sets with positive reach, which form a common extension of closed convex sets and compact $C^2$-domains with boundary,  were introduced  and applied by Federer \cite{Fed59}. Later their natural generalization have been investigated also in Riemannian manifolds (see \cite{Ban82}) and in general Hilbert spaces (see e.g. \cite{CT10,ThII}), under the name \emph{prox-regular sets}. Currently, sets with positive reach are extensively used in data analysis for manifold reconstruction (see e.g.\ \cite{BCM22}).

It appears that sets with positive reach may have much  more complicated geometric or topological  structure than closed convex sets or compact $C^2$-domains. Even the problem of complete
 characterization of the local structure of sets with positive reach in $\mathbb{R}^d$ seems to be too difficult.
However, satisfactory answers are known in some special cases.

The following result   is a consequence of statements claimed in \cite[Remark 4.20]{Fed59}
 and easily follows from \cite[Proposition 7.4]{RZ17}.

\begin{theoremF*} [Federer]
Let  $A \subset \mathbb{R}^d$ be a $k$-dimensional set ($1\leq k<d$) with positive reach, $a \in A$,
and let the tangent cone of $A$ at $a$ be a $k$-dimensional space. Then $A$ agrees on a neighbourhood of $a$ with  a $k$-dimensional $C^{1,1}$ surface.
\end{theoremF*}

By well-known results, if $P$ is a $k$-dimensional $C^{1,1}$ surface (cf.\ Definition~\ref{plochy}), $a\in P$ and $r>0$ is sufficiently
 small, then $P\cap\overline{B}(a,r)$ has positive reach. It follows that  Theorem~F gives a complete satisfactory
 characterization of the local structure of $k$-dimensional sets $A$ with positive reach at a fixed point $a\in\mathbb{R}^d$ such that the tangent cone of $A$ at $a$ is a subspace of dimension $k$: it is
 the same as the local structure of $k$-dimensional $C^{1,1}$ surfaces containing $a$. (More formally, the two systems
 of the corresponding germs coincide.)

Recently, A. Lytchak proved an interesting result (\cite[Theorem 1.2]{Ly24}) which is essentially an extension of Theorem~F. Lytchak's result deals with sets in $d$-dimensional Riemannian manifold; its (essentially equivalent) Euclidean version is the following.

\begin{theoremL*} [Lytchak]
Let  $A \subset \mathbb{R}^d$ be a $k$-dimensional set ($1\leq k\leq d$) with positive reach, $a \in A$, and let the tangent cone of $A$ at $a$ have dimension $k$. Then there exist a convex body $K\subset\mathbb{R}^k\subset \mathbb{R}^d$ (of dimension $k$), an open set $U\subset\mathbb{R}^d$ containing $K$ and a $C^{1,1}$-diffeomorphism  $\Phi: U \to \mathbb{R}^d$  such that $\Phi(K)$ is a neighbourhood of $a$ in $A$.
\end{theoremL*}

Since any set  $\Phi(K)$ as above and containing $a$ has all properties of the set $A$ from Theorem~L, we see that Theorem~L gives a characterization of the local structure of sets $A$ with positive reach at points $a$ where the dimension of the tangent cone is the same as the dimension of $A$ (cf.\ our remark after Theorem~F).

Note that Theorem~L not only generalizes Theorem~F (which corresponds to the case $\Phi^{-1}(a)  \in {\rm int}\, K$), but also gives (for $k=1$) a local characterization of one-dimensional sets $A$ with positive reach in $\mathbb{R}^d$ which is equivalent to that which follows from  \cite[Corollary~8.9]{RZ17}: on a neighbourhood of $a\in A$, $A$ is either a singleton, or a simple $C^{1,1}$ arc.

If $A\subset\mathbb{R}^2$ has positive reach then its local description using semiconcave and semiconvex functions (without any further restrictions) follows easily from \cite[Theorem~6.4]{RZ17}. This result, however, cannot be easily extended to higher dimensions.

The main result of the present article is a complete characterization of the local structure of two-dimensional sets $A$ with positive reach in $\mathbb{R}^d$ for $d\geq 3$. Note that Theorem~L gives such a characterization at points $a\in A$ where $\operatorname{Tan}(A,a)$ has dimension $2$; the only nontrivial remaining case is when $\dim\operatorname{Tan}(A,a)=1$.  This case, however, is much  different from the case $\dim\operatorname{Tan}(A,a)=2$ - in particular, we cannot expect that $A$ will lie on a two-dimensional $C^{1,1}$ surface in some neighbourhood of $a$. This follows already from \cite[Example~7.13]{RZ17}, which we will describe now in a slightly generalized version.

Consider $r>0$ and Lipschitz functions
$\psi \leq 0 \leq \varphi$ on $[0,r]$ such that $\varphi$ is semiconcave on $(0,r)$, $\psi$
is semiconvex on $(0,r)$, $\varphi(0)= \psi(0)=0$, $\varphi'_+(0)= \psi'_+(0)=0$, and set
$B:= \{(x,y):\, x\in [0,r],\,  \psi(x) \leq y \leq \varphi(x)\}$.
Then $B$ has positive reach (see Lemma~\ref{B_set_PR}).
We further perturbe $B\subset\mathbb{R}^2\subset\mathbb{R}^3$ to a set $M\subset \mathbb{R}^3$ by a ``multirotation'' by the following way.
We set $C:= \{x\in [0,r]: \varphi(x)=\psi(x)\}$, denote by $\Lambda$ the set of all components of $(0,r)\setminus C$ and for each $\lambda \in \Lambda$ denote $B_\lambda = \{(x,y) \in B:\ x\in \overline \lambda\}$. Now we rotate each  $B_\lambda$  in $\mathbb{R}^3$ around the $x$-axis by an angle $\theta(\lambda)$ in the positive sense to a set $M_\lambda$ and set
\begin{equation}  \label{M}
    M:= (B\setminus \bigcup _{\lambda \in \Lambda} B_\lambda)
 \cup   \bigcup _{\lambda \in \Lambda} M_\lambda.
\end{equation}
Then  $M$ has positive reach as well (it follows from Lemma~\ref{prorot}) and its tangent cone at $0$ is a half-line. It is easy to see that if $C$ is totally disconnected and $0$ is its accumulation point, we can choose the rotations $M_\lambda$ of $B_\lambda$ so that no neighbourhood of the origin in $M$ lies on a two-dimensional $C^{1,1}$ surface.

Our main result shows (rather surprisingly) that the above example is in a sense universal (up to a $C^{1,1}$-diffeomorphism) for the case when the tangent cone of $A$ at $a$ is a half-line. In our proof of the main result (Theorem~\ref{T_main} below), we need to include the ``both-sided'' version of the set $B$ above (where the tangent cone is the full line, see case (ii) in Definition~\ref{B-sets}), and to define properly the ``multirotations'' in $\mathbb{R}^d$. So, before stating the result, we need two definitions.

\begin{definition}[B-sets]  \label{B-sets}
\begin{enumerate}
    \item[(i)] We call $M\subset \mathbb{R}^2$  a \emph{$\B_-$-set} if there exist $r>0$, Lipschitz functions $\psi \leq 0 \leq \varphi$ on $[0,r]$ such that $\varphi$ is semiconcave on $(0,r)$, $\psi$ is semiconvex on $(0,r)$, $\varphi(0)= \psi(0)=0$, $\varphi'_+(0)= \psi'_+(0)=0$ and $M= \{(x,y):\ x\in [0,r],  \psi(x) \leq y \leq \varphi(x)\}$.
    \item[(ii)] We call $M\subset \mathbb{R}^2$  a \emph{$\B_+$-set} if there exist $r>0$, Lipschitz functions $\psi \leq 0 \leq \varphi$ on $[-r,r]$ such that $\varphi$ is semiconcave on $(-r,r)$, $\psi$ is semiconvex on $(-r,r)$, $\varphi(0)= \psi(0)=0$, $\varphi'(0)= \psi'(0)=0$ and $M= \{(x,y):\ x\in [-r,r],  \psi(x) \leq y \leq \varphi(x)\}$.
    \item[(iii)] We say that $M\subset\mathbb{R}^2$ is a \emph{$\operatorname{B}$-set} if $M$ is a $\B_-$-set or a $\B_+$-set.
\end{enumerate}
\end{definition}

Any $\operatorname{B}$-set has positive reach (see Lemma~\ref{B_set_PR}).

Let $SO(d)$ be the group of rotations in $\mathbb{R}^d$ (i.e., linear mappings preserving the scalar product and orientation). We call an element $R\in SO(d)$ a \emph{2-rotation}
if there exists a two-dimensional subspace $T$ of $\mathbb{R}^d$ orthogonal to $e_1$ and such that $Rv=v$ for any $v\in T^\perp$. (Notice that 2-rotations are determined by rotations in some 2-plane orthogonal to $e_1$.) Note that $R^{-1}$ is a 2-rotation whenever $R$ is.

Let $\pi$ denote the orthogonal projection onto the $x_1$-axis $\operatorname{span}\{e_1\}$.

\begin{definition}[multirotations]  \label{multrot}
Let $\emptyset\neq C\subset\operatorname{span}\{e_1\}\subset\mathbb{R}^d$ ($d\geq 3$) be a compact set and $\Lambda$ the family of all components (open intervals) of $\operatorname{span}\{e_1\}\setminus C$.
We will call a mapping $\rho:\mathbb{R}^d\to\mathbb{R}^d$ \emph{multirotation} (in $\mathbb{R}^d$) \emph{associated with $C$} if
$\rho(x)=x$ whenever $\pi(x)\in C$ and for any $\lambda\in\Lambda$ there exists a 2-rotation $R_\lambda$ such that $\rho|_{\pi^{-1}(\lambda)}=R_\lambda|_{\pi^{-1}(\lambda)}$.
We will say that the multirotation $\rho$ is determined by the family of $2$-rotations $(R_\lambda)_{\lambda\in\Lambda}$.
\end{definition}

Given a set $A\subset\mathbb{R}^d$ with positive reach and $k\in\{0,1,\dots,d\}$, we denote by $T_k(A)$ the set of all points of $A$ at which the tangent cone of $A$ has dimension $k$.

Note that if $B$ is a B-set then, since it has positive reach, $T_1(B)$ is a compact subset of $\mathbb{R}=\operatorname{span}\{e_1\}$ by Lemma~\ref{T_k}~(iii).

\begin{theoremL*} [Lytchak]
Let $A \subset \mathbb{R}^d$ be a $k$-dimensional set ($1\leq k\leq d$) with positive reach, $a \in A$, and let the tangent cone of $A$ at $a$ have dimension $k$. Then there exist a convex body $K\subset\mathbb{R}^k\subset \mathbb{R}^d$ (of dimension $k$), an open set $U\subset\mathbb{R}^d$ containing $K$ and a $C^{1,1}$-diffeomorphism $\Phi: U \to \mathbb{R}^d$ such that $\Phi(K)$ is a neighbourhood of $a$ in $A$.
\end{theoremL*}

Consider $r>0$ and Lipschitz functions
$\psi \leq 0 \leq \varphi$ on $[0,r]$ such that $\varphi$ is semiconcave on $(0,r)$, $\psi$
is semiconvex on $(0,r)$, $\varphi(0)= \psi(0)=0$, $\varphi'_+(0)= \psi'_+(0)=0$, and set
$B:= \{(x,y):\, x\in [0,r],\, \psi(x) \leq y \leq \varphi(x)\}$.
Then $B$ has positive reach (see Lemma~\ref{B_set_PR}).
We further perturbe $B\subset\mathbb{R}^2\subset\mathbb{R}^3$ to a set $M\subset \mathbb{R}^3$ by a ``multirotation'' by the following way.
We set $C:= \{x\in [0,r]: \varphi(x)=\psi(x)\}$, denote by $\Lambda$ the set of all components of $(0,r)\setminus C$ and for each $\lambda \in \Lambda$ denote $B_\lambda = \{(x,y) \in B:\ x\in \overline \lambda\}$. Now we rotate each $B_\lambda$ in $\mathbb{R}^3$ around the $x$-axis by an angle $\theta(\lambda)$ in the positive sense to a set $M_\lambda$ and set
\begin{equation} \label{M}
 M:= (B\setminus \bigcup _{\lambda \in \Lambda} B_\lambda)
 \cup \bigcup _{\lambda \in \Lambda} M_\lambda.
\end{equation}
Then $M$ has positive reach as well (it follows from Lemma~\ref{prorot}) and its tangent cone at $0$ is a half-line. It is easy to see that if $C$ is totally disconnected and $0$ is its accumulation point, we can choose the rotations $M_\lambda$ of $B_\lambda$ so that no neighbourhood of the origin in $M$ lies on a two-dimensional $C^{1,1}$ surface.

\begin{definition}[B-sets] \label{B-sets}
\begin{enumerate}
 \item[(i)] We call $M\subset \mathbb{R}^2$ a \emph{$\B_-$-set} if there exist $r>0$, Lipschitz functions $\psi \leq 0 \leq \varphi$ on $[0,r]$ such that $\varphi$ is semiconcave on $(0,r)$, $\psi$ is semiconvex on $(0,r)$, $\varphi(0)= \psi(0)=0$, $\varphi'_+(0)= \psi'_+(0)=0$ and $M= \{(x,y):\ x\in [0,r], \psi(x) \leq y \leq \varphi(x)\}$.
 \item[(ii)] We call $M\subset \mathbb{R}^2$ a \emph{$\B_+$-set} if there exist $r>0$, Lipschitz functions $\psi \leq 0 \leq \varphi$ on $[-r,r]$ such that $\varphi$ is semiconcave on $(-r,r)$, $\psi$ is semiconvex on $(-r,r)$, $\varphi(0)= \psi(0)=0$, $\varphi'(0)= \psi'(0)=0$ and $M= \{(x,y):\ x\in [-r,r], \psi(x) \leq y \leq \varphi(x)\}$.
 \item[(iii)] We say that $M\subset\mathbb{R}^2$ is a \emph{$\operatorname{B}$-set} if $M$ is a $\B_-$-set or a $\B_+$-set.
\end{enumerate}
\end{definition}

\begin{definition}[multirotations] \label{multrot}
Let $\emptyset\neq C\subset\operatorname{span}\{e_1\}\subset\mathbb{R}^d$ ($d\geq 3$) be a compact set and $\Lambda$ the family of all components (open intervals) of $\operatorname{span}\{e_1\}\setminus C$.
We will call a mapping $\rho:\mathbb{R}^d\to\mathbb{R}^d$ \emph{multirotation} (in $\mathbb{R}^d$) \emph{associated with $C$} if
$\rho(x)=x$ whenever $\pi(x)\in C$ and for any $\lambda\in\Lambda$ there exists a 2-rotation $R_\lambda$ such that $\rho|_{\pi^{-1}(\lambda)}=R_\lambda|_{\pi^{-1}(\lambda)}$.
We will say that the multirotation $\rho$ is determined by the family of $2$-rotations $(R_\lambda)_{\lambda\in\Lambda}$.
\end{definition}

Given a set $A\subset\mathbb{R}^d$ with positive reach and $k\in\{0,1,\dots,d\}$, we denote by $T_k(A)$ the set of all points of $A$ at which the tangent cone of $A$ has dimension $k$.

Note that if $B$ is a B-set then, since it has positive reach, $T_1(B)$ is a compact subset of $\mathbb{R}=\operatorname{span}\{e_1\}$ by Lemma~\ref{T_k}~(iii).

\begin{lemma} \label{B_set_PR}
    Any $\B$-set $B\subset\R^2$ has positive reach.
\end{lemma}

\begin{lemma}   \label{T_k}
    Let $A\subset\R^d$ have positive reach and $1\leq k\leq d$. Then
    \begin{enumerate}
        \item[(i)] $T_k(A)\cap\overline{\bigcup_{i=0}^{k-1}T_i(A)}=\emptyset$,
        \item[(ii)] the mapping $\psi_k^A$ is continuous on $T_k(A)$,
        \item[(iii)] $T_1(A)$ is closed,
        \item[(iv)] $T_0(A)$ is the set of all isolated points of $A$.
    \end{enumerate}
\end{lemma}

\begin{lemma}\label{prorot}
Let $A\subset\R^d$ ($d\geq  3$) be a set with positive reach skewered on a segment $[p,q]\subset \R$ and let $r>0$ be such that  $0< r < \reach A$ and
\begin{enumerate}
    \item[(a)] $\diam A < r$, or
    \item[(b)] $A\subset \R^2 \subset \R^d$ is a $B$-set.
\end{enumerate}
Let $\rho$ be a multirotation associated with $T_1(A)$.
Then
\begin{enumerate}
\item[(i)]
the mapping  $\rho|_A$ is bi-Lipschitz,
\item[(ii)]
 $\rho(A)$ is compact and $\reach \rho(A) > 0$,
\item[(iii)] $T_1(\rho(A))=T_1(A)$ and $\rho(A)$ is skewered on $[p,q]$.
\end{enumerate}
\end{lemma}

Note that if $B$ is a B-set then, since it has positive reach, $T_1(B)$ is a compact subset of $\R=\spa\{e_1\}$ by Lemma~\ref{T_k}~(iii).

\begin{remark}
 It is easy to see that Theorem~\ref{T_main} holds if we replace ``$\dim A\leq 2$'' with ``$\dim A=2$'' at both occurrences. The only difference is in the proof of the ``moreover'' part: While in the first case we set in our proof $A:=\Phi(\rho(B))$, in the latter case we could take $A:=\Phi(\rho(B))\cup D$, where $\reach D>0$, $\dim D=2$ and $\dist(\Phi(\rho(B)),D)>0$.
\end{remark}

\begin{corollary}
 Let $A\subset\R^d$ ($d\leq 3$) be a two-dimensional compact set. Then $A$ has positive reach if and only if for each non-isolated $a\in A$ there exist a compact set $K\subset\R^d$, an open set $K\subset U\subset\R^d$ and a $C^{1,1}$-diffeomorphism $\Phi:U\to\R^d$ such that $\Phi(K)$ is a neighbourhood of $a$ in $A$ and either
 \begin{enumerate}
 \item[(a)] $K\subset\R^2\subset\R^d$ is convex two-dimensional, or
 \item[(b)] $K=\rho(B)$, where $B\subset\R^2\subset\R^d$ is a $\B$-set and $\rho:\R^d\to\R^d$ is a multirotation associated with $T_1(B)$.
 \end{enumerate}
\end{corollary}

\begin{proposition} \label{T}
 Let $A\subset\R^d$ ($d\geq 3$) have positive reach, $\dim A\leq 2$ and $a\in T_1(A)$. Then there exists a $\B$-set $B$ and a multirotation $\rho$ associated with $T_1(B)$ such that $(A,a)\approx(\rho(B),0)$.
\end{proposition}

\begin{proposition} \label{T3}
 Let $S\subset\R^d$ ($d\geq 3$) with positive reach be a set skewered on $[c,d]$, $c<0<d$, with planar leaves, and $0\in T_1(S)$. Then there exists a $\B_+$-set $B$ and a multirotation $\rho$ associated with $T_1(B)$ such that $(S,0)\sim(\rho(B),0)$.
\end{proposition}

Let us consider now the case $a\in T_1^-(A)$. Choose $0<r<\reach A$ and let $u$ be the unit vector such that $\Tan(A,a)=\{tu:\, t\geq 0\}$. Denoting $S:=\{a+tu:\, -\frac r4\leq t< 0\}$ and $A^*:=A\cup S$, we have clearly $a\in T_1^+(A^*)$ and \cite[Lemma 3.8]{RZ17} gives $\reach A^* \geq r/4$. We apply the already proved assertion and find a $\B_+$-set $B^*$ and a multirotation $\rho^*$ associated with $T_1(B^*)$ such that $(A^*,a)\approx(\rho^*(B^*),0)$.
By Remark~\ref{oekv1}~(i) there exist a $C^{1,1}$-diffeomorphism $\Phi$ defined on a neighbourhood $U$ of $a$ and $\ep>0$ such that
\begin{equation} \label{AB}
\Phi(a)=0 \ \text{ and }\ \Phi(A^*\cap U)\cap B(0,\ep)=\rho^*(B^*)\cap B(0,\ep).
\end{equation}
Further, there exists $0<\delta<r/4$ such that
\begin{equation} \label{S_delta}
 B(a,\delta)\subset U\text{ and }\Phi(B(a,\delta))\subset B(0,\ep).
\end{equation}
Denote $S_\delta:=S\cap B(a,\delta)=\{a+tu:\, -\delta< t< 0\}$.
Since $S\subset T_1(A^*)$, we obtain
$$\Phi(S_\delta)\subset \Phi(T_1(A^*)\cap U)\cap B(0,\ep) \subset T_1(\rho^*(B^*))$$
using Lemma~\ref{tkdif} and \eqref{AB}.
We have further $T_1(\rho^*(B^*))=T_1(B^*)\subset W$ by Lemma~\ref{prorot}~(iii) and Remark~\ref{Rem_listky}~(b), hence we get $\Phi(S_\delta)\subset W$. Since $\Phi(S_\delta)$ is homeomorphic to $(0,1)$ and $0\in\overline{\Phi(S_\delta)}\setminus \Phi(S_\delta)$, we have (a) $\Phi(S_\delta)=(-\ep',0)$ or (b) $\Phi(S_\delta)=(0,\ep')$ for some $0<\ep'\leq \ep$.
We claim that we can choose $B^*,\rho^*,\Phi,U,\ep$ and $\delta$ as above satisfying \eqref{AB} and \eqref{S_delta} such that the case (a) occurs. Indeed, if $B^*,\rho^*,\Phi,U,\ep$ and $\delta$ satisfy \eqref{AB}, \eqref{S_delta} and (b), consider the reflection $G:(x,z)\mapsto(-x,z)$ ($(x,z)\in\R\times\R^{d-1}$) and set $\tilde{B}^*:=G(B^*)$, $\tilde{\rho}^*:=G\circ\rho^*\circ G$, $\tilde{\Phi}:=G\circ\Phi$, $\tilde{U}:=U$, $\tilde{\ep}:=\ep$ and $\tilde{\delta}:=\delta$. Then both \eqref{AB} and \eqref{S_delta} remain true and it is easy to see that $\tilde{B}^*$ is again a $\B_+$-set (using Remark~\ref{rem_semicon}) and that $\tilde{\rho}^*$ is a multirotation associated with
 $T_1(\tilde{B}^*)$. Moreover, $\tilde{\Phi}(S_{\tilde{\delta}})=-\Phi(S_\delta)=(-\ep',0)$, i.e., case (a) occurs.

For the second statement of Theorem~\ref{T_main}, assume that $B\subset\R^2\subset\R^d$ is a $\B$-set, $\rho$ a multirotation associated with $T_1(B)$, $U$ an open neighbourhood of $\rho(B)$ and $\Phi:U\to\R^d$ a $C^{1,1}$-diffeomorphism. Then $B$ has positive reach by Lemma~\ref{B_set_PR} and it is clearly compact, hence $\rho(B)$ is compact and has positive reach by Lemma~\ref{prorot}~(ii). Setting $A:=\Phi(\rho(B))$, we have $\reach A>0$ by Lemma~\ref{reach_C11} and $a\in T_1(A)$ by Lemma~\ref{prorot}~(iii) and Lemma~\ref{tkdif}~(iii). This implies the second assertion of Theorem~\ref{T_main} and completes the proof.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{bilip}] \hskip 2mm
 Let $A\subset\R^d$ be a two-dimensional set with positive reach and $a\in A$.
 If $a\in T_0(A)$ then $a$ is an isolated point in $A$ and the assertion is obvious.
 If $a\in T_1(A)$ then we apply Theorem~\ref{T_main} and obtain a $\B$-set $B\subset\R^2$, a multirotation $\rho$ associated with $T_1(B)$, a neighbourhood $U$ of $\rho(B)$ in $\R^d$ and a $C^{1,1}$-diffeomorphism $\Phi:U\to\R^d$ such that $\Phi(\rho(B))$ is a neighbourhood of $a$ in $A$.
 The set $\rho(B)$ is compact by Lemma~\ref{prorot}~(ii), hence $\Phi(\rho(B))$ is compact as well. Since both $\Phi$ and $\Phi^{-1}$ are locally Lipschitz (see Remark~\ref{stability}~(i)), their restrictions to compact sets $\rho(B)$ and $\Phi(\rho(B))$ (respectively) are Lipschitz by \cite[Proposition~A48]{Lee}, i.e., $\Phi$ is bi-Lipschitz on $\rho(B)$.
 Since $\rho$ is bi-Lipschitz on $B$ by Lemma~\ref{prorot}~(i), we can choose $\Psi:=\Phi\circ\rho|_B$ and the assertion follows.

\begin{lemma} \label{B_set_PR}
    Any $\B$-set $B\subset\R^2$ has positive reach.
\end{lemma}

\begin{lemma}   \label{T_k}
    Let $A\subset\R^d$ have positive reach and $1\leq k\leq d$. Then
    \begin{enumerate}
        \item[(i)] $T_k(A)\cap\overline{\bigcup_{i=0}^{k-1}T_i(A)}=\emptyset$,
        \item[(ii)] the mapping $\psi_k^A$ is continuous on $T_k(A)$,
        \item[(iii)] $T_1(A)$ is closed,
        \item[(iv)] $T_0(A)$ is the set of all isolated points of $A$.
    \end{enumerate}
\end{lemma}

\begin{theorem}  \label{T_main}
    Let $A\subset\R^d$ ($d\geq 3$) be a set with positive reach with $\dim A\leq 2$ and let $a\in A$ be such that $\dim(\Tan(A,a))=1$. Then there exist a $\B$-set $B\subset\R^2\subset\R^d$, a multirotation $\rho:\R^d\to\R^d$ associated with $T_1(B)$ and a $C^{1,1}$-diffeomorphism $\Phi:U\subset\R^d\to\R^d$ such that $0\in U$, $a=\Phi(0)$, $\rho(B)\subset U$ and $\Phi(\rho(B))$ is a neighbourhood of $a$ in $A$.

    Moreover, if $\Phi$, $B$ and $\rho$ are as above, then $\Phi(\rho(B))$ agrees on a neighbourhood of $a$ with some set $A\subset\R^d$ with positive reach satisfying $\dim A\leq 2$ and $\dim(\Tan(A,a))=1$.
\end{theorem}

\begin{corollary}  \label{bilip}
    Let $A\subset\R^d$ ($d\geq 3$) be a two-dimensional set with positive reach and $a\in A$. Then there exists a compact set $0\in B\subset\R^2$ with positive reach and a bi-Lipschitz mapping $\Psi:B\to A^*$, where $A^*$ is a neighbourhood of $a$ in $A$.
\end{corollary}

\begin{lemma}\label{prorot}
Let $A\subset\R^d$ ($d\geq  3$) be a set with positive reach skewered on a segment $[p,q]\subset \R$ and let $r>0$ be such that  $0< r < \reach A$ and
\begin{enumerate}
    \item[(a)] $\diam A < r$, or
    \item[(b)] $A\subset \R^2 \subset \R^d$ is a $B$-set.
\end{enumerate}
Let $\rho$ be a multirotation associated with $T_1(A)$.
Then
\begin{enumerate}
\item[(i)]
the mapping  $\rho|_A$ is bi-Lipschitz,
\item[(ii)]
 $\rho(A)$ is compact and $\reach \rho(A) > 0$,
\item[(iii)] $T_1(\rho(A))=T_1(A)$ and $\rho(A)$ is skewered on $[p,q]$.
\end{enumerate}
\end{lemma}

\begin{remark}
   It is easy to see that Theorem~\ref{T_main} holds if we replace ``$\dim A\leq 2$'' with ``$\dim A=2$'' at both occurrences. The only difference is in the proof of the ``moreover'' part: While in the first case we set in our proof  $A:=\Phi(\rho(B))$, in the latter case we could take $A:=\Phi(\rho(B))\cup D$, where ${\rm reach}\, D>0$, $\dim D=2$ and $\mathrm{dist}(\Phi(\rho(B)),D)>0$.
\end{remark}

Note that (ignoring the trivial case when $a$ is an isolated point of $A$), Theorem~\ref{T_main} together with Theorem~L give a complete characterization of the local structure of two-dimensional sets $A\subset\mathbb{R}^d$ with positive reach: For a given point $a\in\mathbb{R}^d$, the set of all germs at $a$ of sets $A$ as above with $a\in A$ is described ``constructively''. Also, both results together imply (using also \eqref{reach_compact} and Lemma~\ref{local}) the following characterization of compact two-dimensional sets with positive reach.

\begin{corollary}
    Let $A\subset\mathbb{R}^d$ ($d\leq 3$) be a two-dimensional compact set. Then $A$ has positive reach if and only if for each non-isolated $a\in A$ there exist a compact set $K\subset\mathbb{R}^d$, an open set $K\subset U\subset\mathbb{R}^d$ and a $C^{1,1}$-diffeomorphism $\Phi:U\to\mathbb{R}^d$ such that $\Phi(K)$ is a neighbourhood of $a$ in $A$ and either
    \begin{enumerate}
        \item[(a)] $K\subset\mathbb{R}^2\subset\mathbb{R}^d$ is convex two-dimensional, or
        \item[(b)] $K=\rho(B)$, where $B\subset\mathbb{R}^2\subset\mathbb{R}^d$ is a $\operatorname{B}$-set and $\rho:\mathbb{R}^d\to\mathbb{R}^d$ is a multirotation associated with $T_1(B)$.
    \end{enumerate}
\end{corollary}

In the situation of Theorem~\ref{T_main} (Theorem~L), we can easily observe that the mapping
$\Psi:=\Phi\circ\rho|_B$ ($\Psi:=\Phi|_K$, respectively) is bi-Lipschitz. Using this, we obtain a positive answer to \cite[Question~1.7]{Ly24}:

\begin{corollary}  \label{bilip}
    Let $A\subset\mathbb{R}^d$ ($d\geq 3$) be a two-dimensional set with positive reach and $a\in A$. Then there exists a compact set $0\in B\subset\mathbb{R}^2$ with positive reach and a bi-Lipschitz mapping $\Psi:B\to A^*$, where $A^*$ is a neighbourhood of $a$ in $A$.
\end{corollary}

The proof of Theorem~\ref{T_main} has two principal ingredients which come from Lytchak's proof of Theorem~L.

First, given a set $A\subset\mathbb{R}^d$ with positive reach and $1\leq k\leq d$, we consider the mapping $\psi_k=\psi_k^A:x\mapsto\operatorname{span}\{\operatorname{Tan}(A,x)\}$ from $T_k(A)$ into the Grassmannian $G(d,k)$ and prove its Lipschitzness on some subsets of $T_k(A)$ under additional assumptions on $A$. Second, using the Lipschitzness of $\psi_2$ on a suitable set, we construct a $C^{1,1}$-diffeomorphism applying Whitney's $C^{1,1}$ extension theorem to a suitably chosen mapping $f:D\subset\mathbb{R}^d\to\mathbb{R}^d$. Here our approach differs from that of Lytchak who also uses the Whitney's $C^{1,1}$  extension, but for a mapping from $\mathbb{R}^k$ to $\mathbb{R}^{d-k}$. For our construction of the function $f$ above, we need some basic facts from the theory of product-integration (see \cite{GJ90}), which is the third principal ingredient of our proof.

When proving the Lipschitzness of $\psi_k$, in contrast to Lytchak who applies the theory of CAT($\kappa$) spaces,
we use a more elementary method based on ``tangential regularity'' of sets with positive reach and some well-known facts on ``fullness'' (very close to the more frequently used ``thickness'') of simplices taken from Whitney \cite{Whi57}.
Some additional arguments are needed to prove Lemma~\ref{lipna2} about Lipschitzness of $\psi_2$ for special two-dimensional sets  which is important in the proof of Theorem~\ref{T_main}.

Using our method we also provide an alternative (more elementary) proof of Theorem~L.

Moreover, our method also yields the local Lipschitzness of $\psi_k$ on $T_k(A)$ for \emph{any} set $A\subset\mathbb{R}^d$ with positive reach, which implies that $T_k(A)$ lies locally on a $k$-dimensional $C^{1,1}$-surface (Theorem~\ref{tdmnapl2}). (Note that for $A$ $k$-dimensional, this follows already from Theorem~L.)
In the case $k=1$ we obtain (using a different method) a stronger version (Theorem~\ref{T1+}) which will be used substantially in the proof of our main result (Theorem~\ref{T_main}).
Theorem~\ref{tdmnapl2} easily implies that for $1\leq k<d$, each $k$-dimensional subset of $\mathbb{R}^d$ with positive reach can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces (see Corollary~\ref{count}), which is new up to our knowledge.

The structure of the article is the following.
In Section~2 (Preliminaries), we fix some basic notation and recall some (mostly well-known) facts about $C^{1,1}$ mappings and sets with positive reach.
Section~3 is devoted to the study of Lipschitzness of the mapping
$\psi_k$.
In Section~4 we prove results on the local structure of $T_k(A)$ (Theorems~\ref{tdmnapl2} and \ref{T1+}) and, as a consequence, Corollary~\ref{count}.
In Section~5 we prove our main results, Theorem~\ref{T_main} and Corollary~\ref{bilip}.
Finally, in Section~6 we present an alternative proof of Theorem~L.

The post-theorem introduction explicitly outlines the proof of Theorem~\ref{T_main} via “two principal ingredients” (adapted from Lytchak’s proof of Theorem~L) plus a “third principal ingredient.”

1. **Lipschitz control of the tangent-span map.** For a set $A\subset\mathbb{R}^d$ with positive reach and $1\le k\le d$, consider
\[
\psi_k=\psi_k^A:x\mapsto \operatorname{span}\{\operatorname{Tan}(A,x)\}\quad \text{from }T_k(A)\text{ into }G(d,k).
\]
They “prove its Lipschitzness on some subsets of $T_k(A)$ under additional assumptions on $A$.” For this, “in contrast to Lytchak who applies the theory of CAT($\kappa$) spaces,” they use “a more elementary method based on ‘tangential regularity’ of sets with positive reach” and facts on “fullness” of simplices from Whitney \cite{Whi57}. They note “additional arguments are needed to prove Lemma~\ref{lipna2} about Lipschitzness of $\psi_2$ for special two-dimensional sets … important in the proof of Theorem~\ref{T_main}.”

2. **Build a $C^{1,1}$-diffeomorphism from Lipschitzness of $\psi_2$.** “Using the Lipschitzness of $\psi_2$ on a suitable set, we construct a $C^{1,1}$-diffeomorphism applying Whitney’s $C^{1,1}$ extension theorem to a suitably chosen mapping $f:D\subset\mathbb{R}^d\to\mathbb{R}^d$.” They emphasize their approach differs from Lytchak’s, who applies Whitney extension “for a mapping from $\mathbb{R}^k$ to $\mathbb{R}^{d-k}$.”

3. **Product-integration input.** “For our construction of the function $f$ above, we need some basic facts from the theory of product-integration (see \cite{GJ90}), which is the third principal ingredient of our proof.”

Additionally, the remark gives a proof detail for the “moreover” part: for $\dim A\le 2$ they “set in our proof $A:=\Phi(\rho(B))$,” while if one assumes $\dim A=2$ they can take $A:=\Phi(\rho(B))\cup D$ with ${\rm reach}\, D>0$, $\dim D=2$, and $\mathrm{dist}(\Phi(\rho(B)),D)>0$.

The post-theorem introduction explicitly outlines the proof of the main theorem via “two principal ingredients” (adapted from Lytchak’s proof of Theorem~L) plus a “third principal ingredient.”

1. **Lipschitz control of the tangent-span map.** For a set $A\subset\mathbb{R}^d$ with positive reach and $1\le k\le d$, consider
\[
\psi_k=\psi_k^A:x\mapsto \operatorname{span}\{\operatorname{Tan}(A,x)\}\quad \text{from }T_k(A)\text{ into }G(d,k).
\]
They “prove its Lipschitzness on some subsets of $T_k(A)$ under additional assumptions on $A$.” For this, “in contrast to Lytchak who applies the theory of CAT($\kappa$) spaces,” they use “a more elementary method based on ‘tangential regularity’ of sets with positive reach” and facts on “fullness” of simplices from Whitney (H. Whitney, *Geometric Integration Theory*, 1957). They note “additional arguments are needed” to obtain the following auxiliary lemma about Lipschitzness of $\psi_2$ for special two-dimensional sets, which is important for establishing the main theorem.

\begin{lemma}\label{lipna2}
\tLet  $A\subset \R^d$, $0<r< \reach A$,
\t$\diam A < r$ and
\t $A \subset T_1^-(A)  \cup   T_2(A)$.
\t Then  $\psi_2:=\psi_2^A$  is $(2^{12}\pi/r)$-Lipschitz on $T_2(A)$.
\end{lemma}

2. **Build a $C^{1,1}$-diffeomorphism from Lipschitzness of $\psi_2$.** “Using the Lipschitzness of $\psi_2$ on a suitable set, we construct a $C^{1,1}$-diffeomorphism applying Whitney’s $C^{1,1}$ extension theorem to a suitably chosen mapping $f:D\subset\mathbb{R}^d\to\mathbb{R}^d$.” They emphasize their approach differs from Lytchak’s, who applies Whitney extension “for a mapping from $\mathbb{R}^k$ to $\mathbb{R}^{d-k}$.”

3. **Product-integration input.** “For our construction of the function $f$ above, we need some basic facts from the theory of product-integration (see G. Gripenberg and S.-O. Londen, *[title not provided]*, 1990), which is the third principal ingredient of our proof.”

Additionally, the remark gives a proof detail for the “moreover” part: for $\dim A\le 2$ they “set in our proof $A:=\Phi(\rho(B))$,” while if one assumes $\dim A=2$ they can take $A:=\Phi(\rho(B))\cup D$ with ${\rm reach}\, D>0$, $\dim D=2$, and $\mathrm{dist}(\Phi(\rho(B)),D)>0$.

Question:
Let \(A\subset \mathbb{R}^d\) with \(d\ge 3\) have positive reach, assume \(\dim A\le 2\), and let \(a\in A\) satisfy \(\dim(\operatorname{Tan}(A,a))=1\). Here a \(\operatorname{B}\)-set \(B\subset \mathbb{R}^2\) means either
\[
B=\{(x,y):x\in[0,r],\ \psi(x)\le y\le \varphi(x)\}
\]
for some \(r>0\) and Lipschitz functions \(\psi\le 0\le \varphi\) on \([0,r]\), with \(\varphi\) semiconcave on \((0,r)\), \(\psi\) semiconvex on \((0,r)\), \(\varphi(0)=\psi(0)=0\), and \(\varphi'_+(0)=\psi'_+(0)=0\); or
\[
B=\{(x,y):x\in[-r,r],\ \psi(x)\le y\le \varphi(x)\}
\]
for some \(r>0\) and Lipschitz functions \(\psi\le 0\le \varphi\) on \([-r,r]\), with \(\varphi\) semiconcave on \((-r,r)\), \(\psi\) semiconvex on \((-r,r)\), \(\varphi(0)=\psi(0)=0\), and \(\varphi'(0)=\psi'(0)=0\). Also, \(T_1(B)\) denotes the set of points of \(B\) where the tangent cone has dimension \(1\). A multirotation associated with \(T_1(B)\) is a map \(\rho:\mathbb{R}^d\to\mathbb{R}^d\) such that, writing \(\pi\) for the orthogonal projection onto \(\operatorname{span}\{e_1\}\), one has \(\rho(x)=x\) whenever \(\pi(x)\in T_1(B)\), and on each connected component \(\lambda\) of \(\operatorname{span}\{e_1\}\setminus T_1(B)\), the restriction \(\rho|_{\pi^{-1}(\lambda)}\) coincides with some fixed 2-rotation. Under these hypotheses, which conclusion holds about the local structure of \(A\) near \(a\)?

There exist a \(\operatorname{B}\)-set \(B\subset \mathbb{R}^2\subset \mathbb{R}^d\), a multirotation \(\rho:\mathbb{R}^d\to\mathbb{R}^d\) associated with \(T_1(B)\), and a \(C^{1,1}\)-diffeomorphism \(\Phi:U\subset \mathbb{R}^d\to \mathbb{R}^d\) from an open set \(U\) such that \(0\in U\), \(\Phi(0)=a\), \(\rho(B)\subset U\), and \(\Phi(\rho(B))\) is a neighbourhood of \(a\) in \(A\). Moreover, for any such \(\Phi\), \(B\), and \(\rho\), there exists a set \(A'\subset \mathbb{R}^d\) of positive reach with \(\dim A'\le 2\) and \(\dim(\operatorname{Tan}(A',a))=1\) such that \(\Phi(\rho(B))\) agrees with \(A'\) on a neighbourhood of \(a\); equivalently, there is a neighbourhood \(V\) of \(a\) with \(V\cap \Phi(\rho(B))=V\cap A'\).

Choices:
B. There exist a \(\operatorname{B}\)-set \(B\subset \mathbb{R}^2\subset \mathbb{R}^d\), a multirotation \(\rho:\mathbb{R}^d\to\mathbb{R}^d\) associated with \(T_1(B)\), and a \(C^{1,1}\)-diffeomorphism \(\Phi:U\subset \mathbb{R}^d\to \mathbb{R}^d\) from an open set \(U\) such that \(0\in U\), \(\Phi(0)=a\), \(\rho(B)\subset U\), and \(\Phi(\rho(B))\) is a neighbourhood of \(a\) in \(A\). Moreover, for any such \(\Phi\), \(B\), and \(\rho\), one may always take \(A'=\Phi(\rho(B))\), so that \(\Phi(\rho(B))\) itself has positive reach, \(\dim \Phi(\rho(B))\le 2\), and \(\dim(\operatorname{Tan}(\Phi(\rho(B)),a))=1\).
C. There exist a \(\operatorname{B}\)-set \(B\subset \mathbb{R}^2\subset \mathbb{R}^d\), a multirotation \(\rho:\mathbb{R}^d\to\mathbb{R}^d\) associated with \(T_1(B)\), and a \(C^{1,1}\)-diffeomorphism \(\Phi:U\subset \mathbb{R}^d\to \mathbb{R}^d\) from an open set \(U\) such that \(0\in U\), \(\Phi(0)=a\), \(\rho(B)\subset U\), and \(\Phi(\rho(B))\subset A\).
D. There exist a \(\operatorname{B}\)-set \(B\subset \mathbb{R}^2\subset \mathbb{R}^d\), a multirotation \(\rho:\mathbb{R}^d\to\mathbb{R}^d\) associated with \(T_1(B)\), and a \(C^1\)-diffeomorphism \(\Phi:U\subset \mathbb{R}^d\to \mathbb{R}^d\) from an open set \(U\) such that \(0\in U\), \(\Phi(0)=a\), \(\rho(B)\subset U\), and \(\Phi(\rho(B))\) is a neighbourhood of \(a\) in \(A\). Moreover, for any such \(\Phi\), \(B\), and \(\rho\), there exists a set \(A'\subset \mathbb{R}^d\) of positive reach with \(\dim A'\le 2\) and \(\dim(\operatorname{Tan}(A',a))=1\) such that \(\Phi(\rho(B))\) agrees with \(A'\) on a neighbourhood of \(a\).
E. There exist a \(\operatorname{B}\)-set \(B\subset \mathbb{R}^2\subset \mathbb{R}^d\), a multirotation \(\rho:\mathbb{R}^d\to\mathbb{R}^d\) associated with \(T_2(B)\), and a \(C^{1,1}\)-diffeomorphism \(\Phi:U\subset \mathbb{R}^d\to \mathbb{R}^d\) from an open set \(U\) such that \(0\in U\), \(\Phi(0)=a\), \(\rho(B)\subset U\), and \(\Phi(\rho(B))\) is a neighbourhood of \(a\) in \(A\). Moreover, for any such \(\Phi\), \(B\), and \(\rho\), there exists a set \(A'\subset \mathbb{R}^d\) of positive reach with \(\dim A'\le 2\) and \(\dim(\operatorname{Tan}(A',a))=1\) such that \(\Phi(\rho(B))\) agrees with \(A'\) on a neighbourhood of \(a\).

moreover-part local-vs-global positive reach of image

dropped neighbourhood equality/converse local-model conclusion

required C^{1,1} regularity from Whitney C^{1,1} construction weakened to C^1

multirotation must be associated with T_1(B), not T_2(B)

The stem gives assumptions and definitions but does not explicitly reveal the correct option. The answer must be identified by comparing subtle differences in the conclusion.

This is essentially a theorem-recall item: the stem sets up the exact hypotheses and asks which existence statement holds, so the correct choice is basically the theorem’s conclusion restated.

There is some reasoning pressure because the options differ in meaningful technical ways (local vs global diffeomorphism, C^{1,1} vs C^1, dependence on T_1(B), weaker vs exact conclusion). However, the task is still mainly precise recognition/recall rather than genuinely generative mathematical reasoning.

The distractors are strong and mathematically plausible: they modify regularity, domain/globality, dependence of the multirotation, or weaken the statement. These reflect realistic failure modes when recalling a technical theorem.

Technically well-constructed with strong distractors and little answer leakage, but it is largely a direct theorem-restatement question rather than a genuine reasoning task.


\label{thm: vis}
Let $n\geq 2$, $k\in\{1,2,\ldots,n-1\}$, and $L\ge1$. Suppose $K\subset \mathbb{R}^n$ is compact. Then, the value function $v$ of \eqref{soc problem} is an upper semicontinuous viscosity solution of the fully nonlinear elliptic partial differential equation
$F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition (see Definition \ref{def:visc} below) where
\begin{equation} \label{def: F}
F(p, M):=\inf\bigg\{\!-\frac{1}{2}\mathrm{tr}(Ma):\, a\succeq 0,\, a p = 0,\,  \lambda_{(n-k)}(a)\geq 1,\, \lambda_{(1)}(a)\leq L\bigg\}.
\end{equation}
Suppose, in addition, that there are $T_\iota\!:\mathbb{R}^n\to\mathbb{R}^n$, $\iota\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\subset \accentset{\circ}{T_\iota(K)}$, for which $\lim_{\iota\downarrow1} T_\iota=I$. (Here, $I$ is the identity map on $\mathbb{R}^n$.) Then, the upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition is unique.

\begin{thm} \label{thm: vis}
Let $n\geq 2$, $k\in\{1,2,\ldots,n-1\}$, and $L\ge1$. Suppose $K\subset \mathbb{R}^n$ is compact. Then, the value function $v$ of \eqref{soc problem} is an upper semicontinuous viscosity solution of the fully nonlinear elliptic partial differential equation
$F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition (see Definition \ref{def:visc} below) where
\begin{equation} \label{def: F}
F(p, M):=\inf\bigg\{\!-\frac{1}{2}\mathrm{tr}(Ma):\, a\succeq 0,\, a p = 0,\,  \lambda_{(n-k)}(a)\geq 1,\, \lambda_{(1)}(a)\leq L\bigg\}.
\end{equation}
Suppose, in addition, that there are $T_\iota\!:\rr^n\to\rr^n$, $\iota\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\subset \accentset{\circ}{T_\iota(K)}$, for which $\lim_{\iota\downarrow1} T_\iota=I$. (Here, $I$ is the identity map on $\rr^n$.) Then, the upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition is unique.
\end{thm}

\begin{equation}\label{n' SDE}
\mathrm{d}X_{[n']}(t) = a(X_{[n']}(t))^{1/2}\,\mathrm{d}{W(t)},
\end{equation}
where $a(y) := I - yy^{\top}/|y|^2$ when $y\neq 0$, $a(0):=I$, and $W$ is an $n'$-dimensional standard Brownian motion. The remaining coordinates of $X$ are chosen to be constant. Then, $a^{1/2}$ is continuous on ${\mathbb R}^{n'} \setminus \{0\}$, and $a(y)^{1/2}y = 0$. Thus, for any $\varepsilon\in(0,|x_{[n']}|)$, by Itô's formula,
\[
|X_{[n']}(t)|^2 = |x_{[n']}|^2 + (n-k)t,\quad t\le\inf\big\{t'\ge0:\,|X_{[n']}(t')|\le\varepsilon\big\}.
\]
Consequently, \eqref{n' SDE} has a global weak solution satisfying $|X_{[n']}(t)|\ge|x_{[n']}|$, $t\ge0$. In addition, $\tau_K = (r^2 - |x|^2) / (n-k)$ almost surely under $\mathrm{P}^{\ast}$.

\medskip

For $x=0$, consider a sequence $(x^m)_{m=1}^\infty$ in $K\backslash\{0\}$ going to $x$. By the compactness of $\mathcal{P}_0$ (Lemma \ref{lemma: compact}), the associated sequence $(\mathrm{P}^{\ast}_{x^m})_{m=1}^\infty$ has a subsequence going to a $\mathrm{P}^{\ast}\in\mathcal{P}_0$. Then, it holds $|X(t)|^2 = (n-k) t$, $t \geq 0$, thus $\tau_K = r^2 / (n-k)$, almost surely under $\mathrm{P}^{\ast}$.
\end{proof}

\subsection{Subsolution property of the value function} \label{subsec: sub}

We are now ready to verify that the value function $v$ of \eqref{soc problem} satisfies Definition \ref{def:visc}(a). Since $v^*=v$ (Proposition \ref{prop: dpp}) and $F_*=F$ (Lemma \ref{compute F}), we may replace $v^*$ by $v$ and $F_*$ by $F$ in Definition \ref{def:visc}(a). Moreover, $v(x)>0$ for all $x\in \accentset{\circ}{K}$ by Example \ref{ex: ball}. Thus, we only need to study $x\in K$ with $v(x)>0$. Let $\varphi\in C^2(\mathbb{R}^n)$ be a test function such that $\varphi\ge v$ on $K$ and $\varphi(x)=v(x)$. Since $\varphi$ can be replaced by $\varphi+|\cdot-x|^4$, we may assume that $\varphi>v$ on $K\backslash\{x\}$.

\medskip

We prove the inequality \eqref{eq: subs ineq} by contradiction. Suppose that $F(\nabla\varphi(x),\nabla^2\varphi(x))>1$. Then, we let
\begin{equation}
\mathcal{A}:=\big\{a\succeq0:\,\lambda_{(n-k)}(a)\ge1,\,\lambda_{(1)}(a)\le L\big\}
\end{equation}

\begin{lemma} \label{lem: convex hull}
Let $n\ge2$ and $k\in\{1,2,\ldots,n-1\}$. Then, the convex hull of the set $\{a \in \mathbb{S}_+^n \!: \lambda_{(n-k)}(a)\geq 1\}$ is
\begin{equation}
\big\{a \in \mathbb{S}^n_+:\,\Pi_m(a)\geq m-k \text{ for }m=k+1,\, k+2,\,\ldots,\, n \big\}=:A.
\end{equation}
\end{lemma}

\label{sec1}

In \cite[Section 3.3]{fernholz_stochastic_2002}, Fernholz has introduced the concept of portfolio domination, now more commonly referred to as \textit{relative arbitrage}, to rigorously capture the notion of ``beating the market''. More specifically, consider a market with $d\ge2$ assets whose vector $\mu:=(\mu_1,\mu_2,\ldots,\mu_d)$ of market weights (i.e., market capitalizations as fractions of the total market capitalization) constitutes a continuous semimartingale with respect to some stochastic basis. A predictable $\mu$-integrable process $\theta$ is called a trading strategy. We refer to $V^{\theta}:=\theta^{\top} \mu := \sum_{i=1}^d \theta_i \,\mu_i$ as the value process of $\theta$ relative to the market and only consider self-financing trading strategies, meaning:
\begin{equation}
V^{\theta}(t) = V^\theta(0)+\int_0^t \theta(s)^{\top}\,\mathrm{d}\mu(s), \quad t\geq 0.
\end{equation}
Taking the market (i.e., the trading strategy $(1,1,\ldots,1)$) as the benchmark, we can state the definition of relative arbitrage from the seminal monograph by Fernholz and Karatzas (\cite[Definition 6.1]{karatzas_stochastic_2009}) as follows.

\begin{defn}
A trading strategy $\theta$ is called a \textit{relative arbitrage} over a time horizon $[0, T]$ if its value process $V^{\theta}$ relative to the market satisfies:
\begin{enumerate}[(a)]
    \item $V^\theta\ge0$ almost surely,
 	\item $V^{\theta}(T) \ge V^{\theta}(0)$ almost surely,
    \item $V^{\theta}(T) > V^{\theta}(0) $ with positive probability.
\end{enumerate}
\end{defn}

\smallskip

The terms `trading strategy', `value process', `self-financing', and `arbitrage' used above
take on their customary meaning when one starts from a market without a bank account and employs the value of the portfolio continuously investing according to the market weights (`market portfolio') as the num\'{e}raire. The latter is emphasized by the term `relative (to the market)'.

\medskip

A central problem of stochastic portfolio theory, going back to the foundational work \cite{fernholz_stochastic_2002} by Fernholz, is to identify classes of markets that admit relative arbitrages over suitable time horizons $[0,T]$. In particular, it is proven in \cite[Example 3.3.3]{fernholz_stochastic_2002} that every market in which the smallest eigenvalue of the underlying assets' instantaneous covariation matrix is bounded away from $0$ and the largest market weight is bounded away from $1$ admits a relative arbitrage over all long enough time horizons $[0,T]$. Surprisingly, the existence of relative arbitrages for the same kind of markets has been shown to hold over \textit{any} non-trivial time horizon in \cite{fernholz_diversity_2005}. This phenomenon, now known as \textit{short-term relative arbitrage}, has been demonstrated in \cite{fernholz_relative_2005} also for the so-called volatility-stabilized markets. We refer further to \cite{KR17}, \cite{vovk18}, \cite{Cu19}, \cite{RX19}, \cite{KK20}, \cite{itkin25} for various extensions of these findings.

\medskip

The practically important distinction between relative arbitrage over long enough time horizons and short-term relative arbitrage naturally prompted the following question (see \cite[Section 4]{BF08}). Does every sufficiently volatile market, in the sense of
\begin{equation}\label{FK cond}
\sum_{i=1}^d  \frac{1}{\mu_i(t)}\,\frac{\mathrm{d}{\langle\mu_i\rangle}(t)}{\mathrm{d}t}  \ge C>0,\quad t\geq 0
\end{equation}
-- a variant of the key assumption behind the construction in \cite[Example 3.3.3]{fernholz_stochastic_2002} highlighted in \cite[Section 3]{fernholz_relative_2005}\footnote{Note that the left-hand side in \eqref{FK cond} is precisely the `excess growth rate' in \cite[display (3.1)]{fernholz_relative_2005}.}, admit a short-term relative arbitrage, and not only relative arbitrages over long enough time horizons? In another surprising twist, this question has been answered negatively in  \cite[Section 6]{fernholz_volatility_2018}. Hence, the focus has shifted to finding the \textit{smallest} $T^*>0$ such that a relative arbitrage over the time horizon $[0,T]$ is possible for any $T>T^*$ in every sufficiently volatile market, usually referred to as the \textit{relative arbitrage problem}.

\medskip

The relative arbitrage problem appears intractable at first glance, but two remarkable insights by Larsson and Ruf (see \cite{larsson_relative_2021}) have allowed them to characterize $T^*$ when the sufficient volatility condition \eqref{FK cond} is replaced by
\begin{equation}\label{LR cond}
\sum_{i=1}^d \langle\mu_i\rangle(t)\ge t,\quad t\geq 0
\end{equation}
(a prominent variant of \eqref{FK cond}, see, e.g., \cite[Example 5.5]{KR17} and apply a time change). Firstly, \cite{larsson_relative_2021} uses the Fundamental Theorem of Asset Pricing to express $T^*$ through the value function of a stochastic optimal control problem. Secondly, \cite{larsson_relative_2021} identifies the Hamilton-Jacobi-Bellman equation associated with the latter as the arrival time formulation of the minimum curvature flow, a geometric flow akin to the celebrated mean curvature flow (see \cite{Mullins56}, \cite{Brakke78}, \cite{Huisken84}), on the probability simplex. The arrival time formulation of the minimum curvature flow on the probability simplex turns out to have a unique viscosity solution that eventually characterizes $T^*$ (see \cite[Theorem~5.1]{larsson_relative_2021}). This resolves the relative arbitrage problem under the sufficient volatility condition \eqref{LR cond}.

\medskip

In the setting of \cite{larsson_relative_2021}, the process of market weights $\mu$
free of relative arbitrage over the time horizon $[0,T^*]$ is expected to have an instantaneous covariation matrix $\big(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\big)_{1\le i,j\le d}$ generically of rank $1$  (cf.~\cite[discussion following Remark 1.4]{larsson_minimum_2022}). At the same time, while instantaneous covariation matrices of large asset universes are commonly estimated with factor models in the empirical finance literature (see \cite[Section 8]{johansson_simple_2023} for a concise overview) and the number of factors used -- equal to the number of dominant eigenvalues reliably estimated -- is indeed much smaller than $d$, this number tends to be much larger than $1$ (e.g., $20$ for $d=238$ in \cite[Figure 8.1]{johansson_simple_2023}). For this reason, we amend the sufficient volatility condition \eqref{LR cond} to
\begin{equation}\label{LSS cond}
\lambda_{(n-k)}\bigg(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le d}\ge 1,\quad \text{a.e. }t\ge0
\end{equation}
where $n:=d-1$, $k$ is any fixed element of $\{1,2,\ldots,n-1\}$, and $\lambda_{(n-k)}$ refers to the $(n-k)$-largest eigenvalue. A situation such as in \cite[Figure 8.1]{johansson_simple_2023} then has $n=237$ and $k=217$ for example. To obtain a positive constant other than $1$ on the right-hand side of~\eqref{LSS cond}, it suffices to apply a simple time change throughout.

\medskip

The ``null hypothesis'' \eqref{LSS cond} is not closed with respect to the convergence in distribution of continuous processes because the lower bound of \eqref{LSS cond} is not preserved under convex combinations of instantaneous covariation matrices. Therefore, we convexify \eqref{LSS cond} (cf. Lemma \ref{lem: convex hull} below):
\begin{equation}\label{LSS cond'}
\Pi_m\bigg(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le d}\geq m-k \quad\text{for}\quad m = k+1,\, k+2,\,\ldots,\, n,\quad \text{a.e. } t\ge0,
\end{equation}
where $\Pi_m(a):= \inf\{\text{tr}(aP)\!: P^2 = P,\,\text{tr}(P)=m\}$ and $\text{tr}$ stands for the trace of a square matrix. Hence, we study the relative arbitrage problem under the sufficient volatility condition \eqref{LSS cond'} together with the technical condition
$\lambda_{(1)}\big(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\big)_{1\le i,j\le d}\le L$, $\text{a.e. }t\ge0$ for some fixed $L\ge1$. The latter may be chosen large enough to accommodate a confidence interval around an empirical estimate of $\lambda_{(1)}\big(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle}{\mathrm{d}t}\big)_{1\le i,j\le d}$. The compactness of the resulting set of instantaneous covariation matrices allows us to establish the semicontinuity of the value function and the dynamic programming principle in the ensuing stochastic optimal control problem.

\medskip

To characterize $T^*$, we begin as in \cite[Section 5]{larsson_relative_2021}. More specifically, we apply a linear transformation $U$ mapping the probability simplex isometrically onto a polytope $K\subset\mathbb{R}^n$. Then, an application of the Fundamental Theorem of Asset Pricing as in \cite[proof of Theorem 3.1]{larsson_relative_2021} yields the representation $T^*=v(U\mu(0))$, where
\begin{equation}\label{soc problem}
v(x) := \sup_{\mathrm{P} \in \mathcal{P}_x} \mathrm{P}\text{-ess} \inf \tau_K\,;
\end{equation}
$\mathcal{P}_x$ is the set of probability measures on $\Omega:=C([0,\infty),\mathbb{R}^n)$, equipped with the Borel $\sigma$-algebra for the topology of locally uniform convergence, under which the coordinate process $X$ is a martingale starting from $x$ and
\begin{equation}\label{eq: constraint}
\begin{split}
& \Pi_m\bigg(\frac{\mathrm{d}\langle X_i,X_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le n}\geq m-k \quad\text{for}\quad m = k+1,\, k+2,\,\ldots,\, n,\quad \text{a.e. }t\ge0, \\
& \lambda_{(1)}\bigg(\frac{\mathrm{d}\langle X_i,X_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le n}\le L,\quad \text{a.e. } t\ge0
\end{split}
\end{equation}
hold almost surely; and
\begin{equation}
\tau_K := \inf\big\{t \geq 0:\, X(t) \notin K\big\}.
\end{equation}
In words: $v(x)$ is the largest, across all martingale laws $\mathrm{P} \in \mathcal{P}_x$, deterministic almost sure lower bound on the exit time from $K$. The following theorem characterizes $v$, for \textit{any compact} $K\subset\mathbb{R}^n$, and is our main result.

The relative arbitrage problem appears intractable at first glance, but two remarkable insights by Larsson and Ruf (see \cite{larsson_relative_2021}) have allowed them to characterize $T^*$ when the sufficient volatility condition \eqref{FK cond} is replaced by
\begin{equation}\label{LR cond}
\sum_{i=1}^d \langle\mu_i\rangle(t)\ge t,\quad t\geq 0
\end{equation}
(a prominent variant of \eqref{FK cond}, see, e.g., \cite[Example 5.5]{KR17} and apply a time change). Firstly, \cite{larsson_relative_2021} uses the Fundamental Theorem of Asset Pricing to express $T^*$ through the value function of a stochastic optimal control problem. Secondly, \cite{larsson_relative_2021} identifies the Hamilton-Jacobi-Bellman equation associated with the latter as the arrival time formulation of the minimum curvature flow, a geometric flow akin to the celebrated mean curvature flow (see \cite{Mullins56}, \cite{Brakke78}, \cite{Huisken84}), on the probability simplex. The arrival time formulation of the minimum curvature flow on the probability simplex turns out to have a unique viscosity solution that eventually characterizes $T^*$ (see \cite[Theorem~5.1]{larsson_relative_2021}). This resolves the relative arbitrage problem under the sufficient volatility condition \eqref{LR cond}.

In the setting of \cite{larsson_relative_2021}, the process of market weights $\mu$
free of relative arbitrage over the time horizon $[0,T^*]$ is expected to have an instantaneous covariation matrix $\big(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\big)_{1\le i,j\le d}$ generically of rank $1$ (cf.~\cite[discussion following Remark 1.4]{larsson_minimum_2022}). At the same time, while instantaneous covariation matrices of large asset universes are commonly estimated with factor models in the empirical finance literature (see \cite[Section 8]{johansson_simple_2023} for a concise overview) and the number of factors used -- equal to the number of dominant eigenvalues reliably estimated -- is indeed much smaller than $d$, this number tends to be much larger than $1$ (e.g., $20$ for $d=238$ in \cite[Figure 8.1]{johansson_simple_2023}). For this reason, we amend the sufficient volatility condition \eqref{LR cond} to
\begin{equation}\label{LSS cond}
\lambda_{(n-k)}\bigg(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le d}\ge 1,\quad \text{a.e. }t\ge0
\end{equation}
where $n:=d-1$, $k$ is any fixed element of $\{1,2,\ldots,n-1\}$, and $\lambda_{(n-k)}$ refers to the $(n-k)$-largest eigenvalue. A situation such as in \cite[Figure 8.1]{johansson_simple_2023} then has $n=237$ and $k=217$ for example. To obtain a positive constant other than $1$ on the right-hand side of~\eqref{LSS cond}, it suffices to apply a simple time change throughout.

The ``null hypothesis'' \eqref{LSS cond} is not closed with respect to the convergence in distribution of continuous processes because the lower bound of \eqref{LSS cond} is not preserved under convex combinations of instantaneous covariation matrices. Therefore, we convexify \eqref{LSS cond} (cf. Lemma \ref{lem: convex hull} below):
\begin{equation}\label{LSS cond'}
\Pi_m\bigg(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le d}\geq m-k \quad\text{for}\quad m = k+1,\, k+2,\,\ldots,\, n,\quad \text{a.e. } t\ge0,
\end{equation}
where $\Pi_m(a):= \inf\{\text{tr}(aP)\!: P^2 = P,\,\text{tr}(P)=m\}$ and $\text{tr}$ stands for the trace of a square matrix. Hence, we study the relative arbitrage problem under the sufficient volatility condition \eqref{LSS cond'} together with the technical condition
$\lambda_{(1)}\big(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle(t)}{\mathrm{d}t}\big)_{1\le i,j\le d}\le L$, $\text{a.e. }t\ge0$ for some fixed $L\ge1$. The latter may be chosen large enough to accommodate a confidence interval around an empirical estimate of $\lambda_{(1)}\big(\frac{\mathrm{d}\langle \mu_i,\mu_j\rangle}{\mathrm{d}t}\big)_{1\le i,j\le d}$. The compactness of the resulting set of instantaneous covariation matrices allows us to establish the semicontinuity of the value function and the dynamic programming principle in the ensuing stochastic optimal control problem.

\medskip

To characterize $T^*$, we begin as in \cite[Section 5]{larsson_relative_2021}. More specifically, we apply a linear transformation $U$ mapping the probability simplex isometrically onto a polytope $K\subset\mathbb{R}^n$. Then, an application of the Fundamental Theorem of Asset Pricing as in \cite[proof of Theorem 3.1]{larsson_relative_2021} yields the representation $T^*=v(U\mu(0))$, where
\begin{equation}\label{soc problem}
v(x) := \sup_{\mathrm{P} \in \mathcal{P}_x} \mathrm{P}\text{-ess} \inf \tau_K\,;
\end{equation}
$\mathcal{P}_x$ is the set of probability measures on $\Omega:=C([0,\infty),\mathbb{R}^n)$, equipped with the Borel $\sigma$-algebra for the topology of locally uniform convergence, under which the coordinate process $X$ is a martingale starting from $x$ and
\begin{equation}\label{eq: constraint}
\begin{split}
& \Pi_m\bigg(\frac{\mathrm{d}\langle X_i,X_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le n}\geq m-k \quad\text{for}\quad m = k+1,\, k+2,\,\ldots,\, n,\quad \text{a.e. }t\ge0, \\
& \lambda_{(1)}\bigg(\frac{\mathrm{d}\langle X_i,X_j\rangle(t)}{\mathrm{d}t}\bigg)_{1\le i,j\le n}\le L,\quad \text{a.e. } t\ge0
\end{split}
\end{equation}
hold almost surely; and
\begin{equation}
\tau_K := \inf\big\{t \geq 0:\, X(t) \notin K\big\}.
\end{equation}
In words: $v(x)$ is the largest, across all martingale laws $\mathrm{P} \in \mathcal{P}_x$, deterministic almost sure lower bound on the exit time from $K$. The following theorem characterizes $v$, for \textit{any compact} $K\subset\mathbb{R}^n$, and is our main result.

\begin{rmk}
\begin{enumerate}[(a)]
\item Theorem \ref{thm: vis} characterizes, in particular, the solution $T^*$ of the described relative arbitrage problem via the representation $T^*=v(U\mu(0))$. \vskip 5.5pt
\item The nonlinearity $F$ is `geometric', as defined in \cite{BSS}, i.e., for any $p \in \rr^n$, symmetric $n\times n$ matrix $M$, $c_1>0$, and $c_2\in\rr$,
\begin{equation}
F(c_1 p, c_1 M + c_2pp^\top) = c_1 F(p,M).
\end{equation}
Parabolic equations with such nonlinearities appear in weak formulations of geometric flows, and the corresponding viscosity theory was first developed in \cite{cgg,evans_motion_1991,son} for the classical mean curvature flow, and then extended in \cite{BSS,ambrosio_level_1996}. \vskip 5.5pt
\item When $L=1$, the partial differential equation $F(\nabla v,\nabla^2 v)=1$ with zero boundary condition becomes the arrival time formulation of a co-dimension mean curvature flow from \cite{ambrosio_level_1996}. For a related but different stochastic representation of these geometric flows we refer to \cite{SoTo}. \vskip 5.5pt
\item In view of the right-hand side of \eqref{soc problem}, it is natural to conjecture that the value function $v$ does not depend on $L$, at least when $K$ is convex. We were not able to show this and leave it as a tantalizing open problem.
\end{enumerate}
\end{rmk}

\begin{prop}\label{prop: dpp}
Let $n\geq 2$, $k\in\{1,2,\ldots,n-1\}$, and $L\ge1$. Suppose $K\subset \mathbb{R}^n$ is compact. Then, the value function $v$ from \eqref{soc problem} is upper semicontinuous on $\mathbb{R}^n$. Moreover, it satisfies the following dynamic programming principle: For any $x\in\mathbb{R}^n$ and any stopping time $\theta$ with respect to the filtration generated by the coordinate process $X$,
\begin{equation}\label{eq: DPP}
v(x) = \sup_{\mathrm{P}\in\mathcal{P}_x} \mathrm{P}\text{-}\mathrm{ess} \inf \, \big(\theta\wedge\tau_K + v(X(\theta))\,\mathbf{1}_{\{\theta\leq \tau_K\}}\big).
\end{equation}
In addition, the supremum in \eqref{eq: DPP} is attained by any optimizer $\mathrm{P}\in \mathcal{P}_x$ in \eqref{soc problem}.
\end{prop}

This section is devoted to verifying that the value function $v$ of \eqref{soc problem} is a viscosity solution to $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition, where
\begin{equation} \label{codim k pde}
F(p, M):=\inf\bigg\{\!-\frac{1}{2}\mathrm{tr}(Ma):\, a\succeq 0,\, a p = 0,\, \lambda_{(n-k)}(a)\geq 1,\, \lambda_{(1)}(a)\leq L\bigg\}.
\end{equation}

\begin{defn} \label{def:visc}
Let $n\geq 2$, $k\in\{1,2,\ldots,n-1\}$, $L\ge1$, and $K\subset \mathbb{R}^n$ be compact.
\begin{enumerate}[(a)]
\item A bounded function $u\!: K \to \mathbb{R}$ is a \textit{viscosity subsolution} of $F(\nabla u, \nabla^2 u)=1$ in $\accentset{\circ}{K}$ if for any $x\in\accentset{\circ}{K}$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u^{\ast}-\varphi)(x)=\max_K (u^{\ast}-\varphi)$, it holds
\begin{equation}\label{eq: subs ineq}
F_{\ast}(\nabla \varphi(x), \nabla^2 \varphi(x))\leq 1.
\end{equation}
The function $u$ satisfies the \textit{zero boundary condition} if for any $x\in\partial K$ with $u^{\ast}(x)>0$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u^{\ast}-\varphi)(x)=\max_K (u^{\ast}-\varphi)$, one has the inequality \eqref{eq: subs ineq}. \\
\item A bounded function $u\!: K \to \mathbb{R}$ is a \textit{viscosity supersolution} of $F(\nabla u, \nabla^2 u)=1$ in $\accentset{\circ}{K}$ if for any $x\in\accentset{\circ}{K}$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u_{\ast}-\varphi)(x)=\min_K (u_{\ast}-\varphi)$, it holds
\begin{equation}\label{eq: supers ineq}
F^{\ast}(\nabla \varphi(x), \nabla^2 \varphi(x))\geq 1.
\end{equation}
The function $u$ satisfies the \textit{zero boundary condition} if for any $x\in\partial K$ with $u_{\ast}(x)<0$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u_{\ast}-\varphi)(x)=\min_K (u_{\ast}-\varphi)$, one has the inequality \eqref{eq: supers ineq}. \\
\item A bounded function $u\!: K \to \mathbb{R}$ is a \textit{viscosity solution} of $F(\nabla u, \nabla^2 u)=1$ on $K$ with \textit{zero boundary condition} if it is a viscosity subsolution in $\accentset{\circ}{K}$ satisfying the zero boundary condition and a viscosity supersolution in $\accentset{\circ}{K}$ satisfying the zero boundary condition.
\end{enumerate}
\end{defn}

\begin{prop} \label{prop: uniq}
In the setting of Theorem \ref{thm: vis}, suppose that there are $T_\iota\!:\rr^n\to\rr^n$, $\iota\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\subset \accentset{\circ}{T_\iota(K)}$, for which $\lim_{\iota\downarrow1} T_\iota=I$. Then, the upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2v)=1$ on $K$ with zero boundary condition is unique.
\end{prop}

\begin{thm}[Comparison Principle] \label{thm: comparison}
In the setting of Theorem \ref{thm: vis}, suppose that there are $T_\iota\!:\rr^n\to\rr^n$, $\iota\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\subset \accentset{\circ}{T_\iota(K)}$, for which $\lim_{\iota\downarrow1} T_\iota=I$. Then, for any upper semicontinuous viscosity subsolution $u$ of $F(\nabla u,\nabla^2u)=1$ on $K$ satisfying the zero boundary condition and any lower semicontinuous viscosity supersolution $w$ of $F(\nabla w,\nabla^2w)=1$ on $K$ satisfying the zero boundary condition, it holds $u\le w^*$.
\end{thm}

\noindent\textbf{Proof.} Consider $w^\iota\!:T_\iota(K)\to\rr$, $x\mapsto w(T_\iota^{-1}x)$. We aim to show that $c_\iota^2 w^\iota$ is a lower semicontinuous viscosity supersolution of $F(\nabla w,\nabla^2w)=1$ on $T_\iota(K)$ satisfying the zero boundary condition, where $c_\iota>0$ is the dilation factor in $T_\iota$. To this end, we claim that for any orthogonal $n\times n$ matrix $O$ and for all $(p,M)\in\rr^n\times\mathbb{S}^n$:
\begin{equation} \label{F transf}
F(p, M) = c_\iota^2\, F(O^\top p,\, c_\iota^{-2} O^{\top} M O).
\end{equation}
Indeed, let $\overline{a}\succeq 0$ with $\overline{a}O^\top p=0$,
$\lambda_{(n-k)}(\overline{a})\geq 1$ and $\lambda_{(1)}(\overline{a})\leq L$. Then, $a:=O\overline{a}O^\top\succeq0$ satisfies $ap=0$, $\lambda_{(n-k)}(a)\geq 1$ and $\lambda_{(1)}(a)\leq L$. Thus, the definition of $F$ in \eqref{def: F} yields
\begin{equation} \label{F transf ineq}
F(p, M) \le - \frac{1}{2} \mathrm{tr}(Ma) = - \frac{c_\iota^2}{2}\,\mathrm{tr}(c_\iota^{-2}O^\top M O\overline{a}).
\end{equation}
Taking the infimum over all $\overline{a}$ as described, we arrive at \eqref{F transf} with ``$\le$''. Conversely, we can use the obtained inequality with $c_\iota$, $O$, and $(p,M)$ replaced by $c_\iota^{-1}$, $O^\top$, and $(O^{\top} p,\, c_\iota^{-2} O^{\top} M O)$, respectively, to find that
\begin{equation}
F(O^{\top} p,\, c_\iota^{-2} O^{\top} M O)
\le c_\iota^{-2}\, F(O O^{\top} p,\,
c_\iota^2 Oc_\iota^{-2} O^{\top} M O O^\top)=c_\iota^{-2}\, F(p, M).
\end{equation}

\begin{rmk}
\begin{enumerate}[(a)]
\item Theorem \ref{thm: vis} characterizes, in particular, the solution $T^*$ of the described relative arbitrage problem via the representation $T^*=v(U\mu(0))$. \vskip 5.5pt
\item The nonlinearity $F$ is `geometric', as defined in \cite{BSS}, i.e., for any $p \in \mathbb{R}^n$, symmetric $n\times n$ matrix $M$, $c_1>0$, and $c_2\in\mathbb{R}$,
\begin{equation}
F(c_1 p, c_1 M + c_2pp^\top) = c_1 F(p,M).
\end{equation}
Parabolic equations with such nonlinearities appear in weak formulations of geometric flows, and the corresponding viscosity theory was first developed in \cite{cgg,evans_motion_1991,son} for the classical mean curvature flow, and then extended in \cite{BSS,ambrosio_level_1996}. \vskip 5.5pt
\item When $L=1$, the partial differential equation $F(\nabla v,\nabla^2 v)=1$ with zero boundary condition becomes the arrival time formulation of a co-dimension mean curvature flow from \cite{ambrosio_level_1996}. For a related but different stochastic representation of these geometric flows we refer to \cite{SoTo}. \vskip 5.5pt
\item In view of the right-hand side of \eqref{soc problem}, it is natural to conjecture that the value function $v$ does not depend on $L$, at least when $K$ is convex. We were not able to show this and leave it as a tantalizing open problem.
\end{enumerate}
\end{rmk}

The remainder of the paper is structured as follows. In Section \ref{sec2}, we show the upper semicontinuity of the value function $v$, as well as a dynamic programming principle it satisfies. In Subsections \ref{subsec: sub} and \ref{subsec super} of Section \ref{sec3}, we establish the viscosity subsolution and supersolution properties of $v$, respectively. Section \ref{sec4} is then devoted to the uniqueness of the upper semicontinuous viscosity solution under the additional assumption in Theorem~\ref{thm: vis}, finishing the proof of the latter. Finally, Section \ref{sec5} examines the continuity of the value function $v$, particularly in the case that $K$ is a polytope as in the setting of stochastic portfolio theory.

\medskip

\noindent\textbf{Acknowledgement.} The authors would like to thank Martin Larsson and Johannes Ruf for many enlightening discussions on the subject of the paper.

In view of the paper structure described after Theorem~\ref{thm: vis}, the proof proceeds as follows: (i) in Section~\ref{sec2} the authors "show the upper semicontinuity of the value function $v$, as well as a dynamic programming principle it satisfies"; (ii) in Section~\ref{sec3} they "establish the viscosity subsolution and supersolution properties of $v$" in Subsections~\ref{subsec: sub} and~\ref{subsec super}, respectively; (iii) Section~\ref{sec4} proves "the uniqueness of the upper semicontinuous viscosity solution under the additional assumption in Theorem~\ref{thm: vis}, finishing the proof of the latter."

In view of the paper structure described after Theorem~\ref{thm: vis}, the proof proceeds as follows: (i) next the authors show the upper semicontinuity of the value function $v$, as well as a dynamic programming principle it satisfies; (ii) they then establish the viscosity subsolution and supersolution properties of $v$ in the following two subsections.

\begin{equation}\label{n' SDE}
\mathrm{d}X_{[n']}(t) = a(X_{[n']}(t))^{1/2}\,\mathrm{d}{W(t)},
\end{equation}
where $a(y) := I - yy^{\top}/|y|^2$ when $y\neq 0$, $a(0):=I$, and $W$ is an $n'$-dimensional standard Brownian motion. The remaining coordinates of $X$ are chosen to be constant. Then, $a^{1/2}$ is continuous on ${\mathbb R}^{n'} \setminus \{0\}$, and $a(y)^{1/2}y = 0$. Thus, for any $\varepsilon\in(0,|x_{[n']}|)$, by Itf4's formula,
\[
|X_{[n']}(t)|^2 = |x_{[n']}|^2 + (n-k)t,\quad t\le\inf\big\{t'\ge0:\,|X_{[n']}(t')|\le\varepsilon\big\}.
\]
Consequently, \eqref{n' SDE} has a global weak solution satisfying $|X_{[n']}(t)|\ge|x_{[n']}|$, $t\ge0$. In addition, $\tau_K = (r^2 - |x|^2) / (n-k)$ almost surely under $\mathrm{P}^{\ast}$.

\medskip

For $x=0$, consider a sequence $(x^m)_{m=1}^\infty$ in $K\backslash\{0\}$ going to $x$. By the compactness of $\mathcal{P}_0$ (Lemma \ref{lemma: compact}), the associated sequence $(\mathrm{P}^{\ast}_{x^m})_{m=1}^\infty$ has a subsequence going to a $\mathrm{P}^{\ast}\in\mathcal{P}_0$. Then, it holds $|X(t)|^2 = (n-k) t$, $t \geq 0$, thus $\tau_K = r^2 / (n-k)$, almost surely under $\mathrm{P}^{\ast}$.
\end{proof}

\subsection{Subsolution property of the value function} \label{subsec: sub}

We are now ready to verify that the value function $v$ of \eqref{soc problem} satisfies Definition \ref{def:visc}(a). Since $v^*=v$ (Proposition \ref{prop: dpp}) and $F_*=F$ (Lemma \ref{compute F}), we may replace $v^*$ by $v$ and $F_*$ by $F$ in Definition \ref{def:visc}(a). Moreover, $v(x)>0$ for all $x\in \accentset{\circ}{K}$ by Example \ref{ex: ball}. Thus, we only need to study $x\in K$ with $v(x)>0$. Let $\varphi\in C^2(\mathbb{R}^n)$ be a test function such that $\varphi\ge v$ on $K$ and $\varphi(x)=v(x)$. Since $\varphi$ can be replaced by $\varphi+|\cdot-x|^4$, we may assume that $\varphi>v$ on $K\backslash\{x\}$.

\medskip

We prove the inequality \eqref{eq: subs ineq} by contradiction. Suppose that $F(\nabla\varphi(x),\nabla^2\varphi(x))>1$. Then, we let
\begin{equation}
\mathcal{A}:=\big\{a\succeq0:\,\lambda_{(n-k)}(a)\ge1,\,\lambda_{(1)}(a)\le L\big\}
\end{equation}

\begin{equation}
0 < \delta \mathbb{E}\left[Z(\theta)\,\mathbf{1}_{\{X(\theta) \in \partial B_{\varepsilon}(x)\cap K\}} \right] \leq \mathbb{E}\left[ Z(\theta) \int_{0}^{\theta} \nabla\varphi(X(s))^{\top} \,\mathrm{d}\widetilde{X}(s) \right] \leq 0,
\end{equation}
a contradiction. The proof of the subsolution property is complete. \ep

\subsection{Supersolution property of the value function} \label{subsec super}

Since $v\ge 0$, it suffices to check the supersolution inequality \eqref{eq: supers ineq} for $x\in\accentset{\circ}{K}$. Fix any $x\in\accentset{\circ}{K}$, and let $\varphi\in C^2(\mathbb{R}^n)$ satisfy $\varphi \le v_*$ on $K$ and $\varphi(x)=v_*(x)$. Since we can study $\varphi-\varepsilon|\cdot-x|^2$ and then pass to the limit $\varepsilon\downarrow0$, we may assume  $\varphi<v_*$ on $K\backslash\{x\}$ and that $\nabla^2 \varphi(x)$ is non-singular. We distinguish two cases: $\nabla\varphi(x)\neq0$ and $\nabla\varphi(x)=0$.

\medskip

\noindent\textbf{Case 1: $\nabla\varphi(x)\neq 0$.} In this case, $F^*(\nabla\varphi(x),\nabla^2\varphi(x))=F(\nabla\varphi(x),\nabla^2\varphi(x))$. We argue by contradiction and suppose that $F(\nabla \varphi(x),\nabla^2\varphi(x))<1$. Then, there exists an $a\in{\mathcal A}$ with
\begin{equation}\label{case 1 a cond}
1+\frac{1}{2}\text{tr}\big(a\nabla^2\varphi(x)\big)>0\quad\text{and}\quad a\nabla\varphi(x) = 0.
\end{equation}

(iii) Finally, under the additional assumption stated in the main theorem, they prove uniqueness of the upper semicontinuous viscosity solution, thereby completing the proof of the main theorem.

\label{thm: vis}
Let $n\geq 2$, $k\in\{1,2,\ldots,n-1\}$, and $L\ge1$. Suppose $K\subset \mathbb{R}^n$ is compact. Then, the value function $v$ of
\begin{equation}\label{soc problem}
v(x) := \sup_{\mathrm{P} \in \mathcal{P}_x} \mathrm{P}\text{-ess} \inf \tau_K\,;
\end{equation}
is an upper semicontinuous viscosity solution of the fully nonlinear elliptic partial differential equation
$F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition, in the following sense.

\begin{defn} \label{def:visc}
Let $n\geq 2$, $k\in\{1,2,\ldots,n-1\}$, $L\ge1$, and $K\subset \mathbb{R}^n$ be compact.
\begin{enumerate}[(a)]
\item A bounded function $u\!: K \to \mathbb{R}$ is a \textit{viscosity subsolution} of $F(\nabla u, \nabla^2 u)=1$ in $\accentset{\circ}{K}$ if for any $x\in\accentset{\circ}{K}$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u^{\ast}-\varphi)(x)=\max_K (u^{\ast}-\varphi)$, it holds
\begin{equation}\label{eq: subs ineq}
F_{\ast}(\nabla \varphi(x), \nabla^2 \varphi(x))\leq 1.
\end{equation}
The function $u$ satisfies the \textit{zero boundary condition} if for any $x\in\partial K$ with $u^{\ast}(x)>0$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u^{\ast}-\varphi)(x)=\max_K (u^{\ast}-\varphi)$, one has the inequality \eqref{eq: subs ineq}. \\
\item A bounded function $u\!: K \to \mathbb{R}$ is a \textit{viscosity supersolution} of $F(\nabla u, \nabla^2 u)=1$ in $\accentset{\circ}{K}$ if for any $x\in\accentset{\circ}{K}$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u_{\ast}-\varphi)(x)=\min_K (u_{\ast}-\varphi)$, it holds
\begin{equation}\label{eq: supers ineq}
F^{\ast}(\nabla \varphi(x), \nabla^2 \varphi(x))\geq 1.
\end{equation}
The function $u$ satisfies the \textit{zero boundary condition} if for any $x\in\partial K$ with $u_{\ast}(x)<0$ and $\varphi\in C^2(\mathbb{R}^n)$ such that $(u_{\ast}-\varphi)(x)=\min_K (u_{\ast}-\varphi)$, one has the inequality \eqref{eq: supers ineq}. \\
\item A bounded function $u\!: K \to \mathbb{R}$ is a \textit{viscosity solution} of $F(\nabla u, \nabla^2 u)=1$ on $K$ with \textit{zero boundary condition} if it is a viscosity subsolution in $\accentset{\circ}{K}$ satisfying the zero boundary condition and a viscosity supersolution in $\accentset{\circ}{K}$ satisfying the zero boundary condition.
\end{enumerate}
\end{defn}

Here
\begin{equation} \label{def: F}
F(p, M):=\inf\bigg\{\!-\frac{1}{2}\mathrm{tr}(Ma):\, a\succeq 0,\, a p = 0,\,  \lambda_{(n-k)}(a)\geq 1,\, \lambda_{(1)}(a)\leq L\bigg\}.
\end{equation}
Suppose, in addition, that there are $T_\iota\!:\mathbb{R}^n\to\mathbb{R}^n$, $\iota\in(1,2]$, each given by a composition of a rotation, a dilation and a translation, and satisfying $K\subset \accentset{\circ}{T_\iota(K)}$, for which $\lim_{\iota\downarrow1} T_\iota=I$. (Here, $I$ is the identity map on $\mathbb{R}^n$.) Then, the upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition is unique.,

Question:
Let $n\ge 2$, $k\in\{1,\dots,n-1\}$, $L\ge 1$, and let $K\subset\mathbb{R}^n$ be compact. For $p\in\mathbb{R}^n$ and $M\in\mathbb{S}^n$, define
\[
F(p,M):=\inf\left\{-\frac12\operatorname{tr}(Ma):\ a\succeq 0,\ ap=0,\ \lambda_{(n-k)}(a)\ge 1,\ \lambda_{(1)}(a)\le L\right\},
\]
where $\lambda_{(j)}(a)$ denotes the $j$-th largest eigenvalue of $a$. Let
\[
v(x):=\sup_{\mathrm P\in\mathcal P_x}\ \mathrm P\text{-ess}\inf\tau_K.
\]
Assume additionally that for each $\iota\in(1,2]$ there is a map $T_\iota:\mathbb{R}^n\to\mathbb{R}^n$, obtained by composing a rotation, a dilation, and a translation, such that $K\subset \operatorname{int}(T_\iota(K))$ and $T_\iota\to I$ as $\iota\downarrow 1$, where $I$ is the identity map. Which statement holds about existence and uniqueness for the problem $F(\nabla u,\nabla^2u)=1$ on $K$ with zero boundary condition in the viscosity sense?

The value function $v$ is an upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition, and it is the unique upper semicontinuous viscosity solution of that boundary value problem on $K$.

Choices:
B. The value function $v$ is a continuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $\operatorname{int}(K)$ with the classical boundary condition $v=0$ on $\partial K$, and under the stated assumption on $T_\iota$ it is the unique continuous viscosity solution of this boundary value problem.
C. The value function $v$ is an upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition.
D. The value function $v$ is an upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition, and under the stated assumption on $T_\iota$ it is the unique viscosity solution of that boundary value problem among all bounded viscosity solutions on $K$.
E. The value function $v$ is an upper semicontinuous viscosity solution of $F(\nabla v,\nabla^2 v)=1$ on $K$ with zero boundary condition, and the same conclusion remains valid without the assumption that $K\subset \operatorname{int}(T_\iota(K))$ and $T_\iota\to I$ as $\iota\downarrow1$.

The stem does not explicitly reveal the correct option. It gives the full PDE setup and hypotheses, but the correct conclusion is not stated outright; the test-taker still has to distinguish among several nearby existence/uniqueness claims.

This is close to theorem-statement recall: the hypotheses and conclusion align very directly with a specific result. However, it is not a pure verbatim restatement, since the options differ in regularity, scope of uniqueness, and necessity of the geometric assumption.

The item requires some reasoning about subtle distinctions (upper semicontinuous vs continuous solutions, uniqueness class, and role of the dilation/rotation hypothesis), but it mainly tests recognition of the exact theorem rather than deeper derivation or synthesis.

Most distractors are mathematically plausible and target common overstatements, such as demanding continuity or dropping a needed hypothesis. But choice C is a weaker true statement implied by A, so the options are not fully mutually exclusive, which weakens distractor quality.

A reasonably well-targeted theorem-recognition MCQ with plausible distractors, but it is somewhat theorem-restatement-heavy and weakened by the inclusion of a weaker true option alongside the intended correct answer.