{\bf Approximation of quasihyperbolic geodesics.}
In spite of all these applications, finding the explicit value of the quasihyperbolic distance is not possible
except in a few special cases.  Therefore, during the past five decades many comparison metrics  have been introduced  that are easily evaluated and, at the same time,  give either upper or lower bounds for the quasihyperbolic metric.  For a survey of these comparison metrics, see \cite{hkv} and the recent  article \cite{mo}.

The problem of finding a shortest path between two given locations is a classical problem in geodesy (see e.g. \cite[Section 13.6]{VermeerRasila00}), and it has many connections to real world questions such as route planning, logistics, robotics, aviation, engineering, and communication networks. In this paper, we investigate this problem for hyperbolic type weighted metrics (see \cite{hkv}). One of the methods to study these problems is to discretize the underlying mathematical model by using graphs. The vertices of the graph are possible intermediate locations between initial and terminal locations, whereas the edges depict all the connections between vertices. The research literature of this part of mathematical graph theory is vast due to the possible savings as a result of process speedup of algorithms.

We study this shortest path problem in a planar domain with a given metric with some
special features: in particular, this metric measures not only the distance between two points
but also their location with respect to the boundary of the domain. In the cases we study, the
existence of shortest paths between two given points, i.e. length-minimizing curves or geodesics, is known by theory. A characteristic feature of the geodesics of hyperbolic type metrics is that they have a tendency to avoid the boundary of the domain as much as possible.

\subsection{Weighted metrics}
For a domain $G \subsetneq \mathbb{R}^n, n\ge 2,$ define a continuous function $w: G \to (0,\infty),$ called {\it weight function} and the {\it weighted
length} of a rectifiable curve $\gamma$ in $ G$ by
\begin{equation}
\label{w-len}
\ell_G^w(\gamma) = \int_{\gamma} \,w(z)\, |dz|\,.
\end{equation}
If $w(x)=1$ for all $x \in G\,,$ then $\ell_G^w(\gamma)$ reduces to the Euclidean length of
$\gamma$, and if
\begin{equation} \label{qh-len}
w(z)= \frac{1}{d(z, \partial G)}\,,
\end{equation}
where $d(z, \partial G)$ is the distance of $z\in G$ from the boundary of the domain, then $\ell_G^w(\gamma)$ is the {\it quasihyperbolic
length} of $\gamma\,.$
The quasihyperbolic distance
 is defined for $x,y \in G$ as
\begin{equation}\label{qhdef}
k_G(x,y)= \inf_{\gamma} \int_{\gamma} \, \frac{|dz|}{d(z, \partial G)}\,.
\end{equation}
where the infimum is taken over all rectifiable curves $\gamma$ in $G$ with
$x,y \in \gamma.$ This metric was defined by F.W. Gehring and his coauthors
B.P. Palka \cite{gp} and B. Osgood \cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \eqref{qhdef} defines a metric and that there exists a length
minimizing curve, geodesic, joining the points $x$ and $y\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \cite{RasilaTalponen14}).

Since its introduction, the quasihyperbolic metric has become a standard tool in the study
of geometric function theory, quasiconformal mappings, and analysis on metric spaces \cite{gh, gmp, hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \cite{b} at most by a constant factor \cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.

\subsection{Weighted metrics}
For a domain $G \subsetneq \mathbb{R}^n, n\ge 2,$ define a continuous function $w: G \to (0,\infty),$ called {\it weight function} and the {\it weighted
length} of a rectifiable curve $\gamma$ in $ G$ by
\begin{equation}
\label{w-len}
\ell_G^w(\gamma) = \int_{\gamma} \,w(z)\, |dz|\,.
\end{equation}
If $w(x)=1$ for all $x \in G\,,$ then $\ell_G^w(\gamma)$ reduces to the Euclidean length of
$\gamma$, and if
\begin{equation} \label{qh-len}
w(z)= \frac{1}{d(z, \partial G)}\,,
\end{equation}
where $d(z, \partial G)$ is the distance of $z\in G$ from the boundary of the domain, then $\ell_G^w(\gamma)$ is the {\it quasihyperbolic
length} of $\gamma\,.$
The quasihyperbolic distance
 is defined for $x,y \in G$ as
\begin{equation}\label{qhdef}
k_G(x,y)= \inf_{\gamma} \int_{\gamma} \, \frac{|dz|}{d(z, \partial G)}\,.
\end{equation}
where the infimum is taken over all rectifiable curves $\gamma$ in $G$ with
$x,y \in \gamma.$ This metric was defined by F.W. Gehring and his coauthors
B.P. Palka \cite{gp} and B. Osgood \cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \eqref{qhdef} defines a metric and that there exists a length
minimizing curve, geodesic, joining the points $x$ and $y\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \cite{RasilaTalponen14}).

Since its introduction, the quasihyperbolic metric has become a standard tool in the study
of geometric function theory, quasiconformal mappings, and analysis on metric spaces \cite{gh, gmp, hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \cite{b} at most by a constant factor \cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.

In this study, quasihyperbolic geometry is investigated from the point of view of constructive approximation.
We discretize the problem by forming a grid of points in the domain $G\,.$
This grid must be fine enough considering the mutual locations of the points
and the geometric features of the domain $G\,.$ We define a graph using
these grid points and define the distance between two neighbouring points
in a manner that is compatible with the quasihyperbolic geometry. The main
step of the process is to find the shortest path in this graph. For this purpose
we use Dijkstra's algorithm \cite{Dijkstra59}. This method applies not only for the
quasihyperbolic metric but also for many other weighted metrics for which shortest
paths joining the points are known to exist. The method enables us to approximate the
quasihyperbolic geodesics by means of polygonal curves and present figures of these approximations in several polygonal
domains. We also compare our approximate values
of the quasihyperbolic distances to approximations
obtained by other methods.

After this introduction the contents of this paper is organized as follows. In Section 2 we introduce some basic notation for the metrics we use, describe how to proceed from grid points of a domain to
a graph, and some facts about special functions. In Section 3 we discuss computing distances in graphs and introduce Dijkstra's algorithm and discuss implementation of the algorithm for approximation of geodesics based on Dijkstra's algorithm. In Section 4 we use this algorithm for approximation of hyperbolic and quasihyperbolic geodesics in the unit disk, with an aim of validating the method and understanding its accuracy. We also consider conformal mappings of the upper half plane onto two polygonal domains, expressed in terms of Schwarz-Christoffel integrals. In Section 5 we investigate cases where non-uniqueness and bifurcation of geodesics is expected from theoretical considerations. In Sections 6--8 we consider other interesting examples where hyperbolic and quasihyperbolic geodesics can be approximated by using our methods. Finally, in Section 9, we give our final conclusions.
\section{Preliminaries}
In this section, we give the formulas and results needed later in our examples. We first focus on hyperbolic geometry and then present the necessary tools in order to handle conformal mappings of the upper half plane onto a quadrilateral domain.

\begin{nonsec}{\bf Distance ratio metric.}
 A useful comparison
metric for the quasihyperbolic metric is the distance ratio metric \cite{gp}
\begin{equation}\label{jdef}
j_G(x,y)= \log \left( 1+ \frac{|x-y|}{\min\{ d(x, \partial G), d(y, \partial G)\}}\right) \,,
\end{equation}
where $x,y \in G$.
The metrics $j_G$ and $k_G$ are not conformally invariant, and they are only
similarity invariant.
If $G \in \{ \mathbb{B}^2, \mathbb{H}^2\}, $ then for
$x,y \in G$
 \begin{equation} \label{jrho}
 j_G(x,y) \le \rho_G(x,y) \le 2 j_G(x,y) \,,
 \end{equation}
 \begin{equation} \label{krho}
 k_G(x,y) \le \rho_G(x,y) \le 2 k_G(x,y) \,.
 \end{equation}
In fact, $k_G \equiv \rho_G $ for $G= \mathbb{H}^2.$
The geodesics of the quasihyperbolic metric are smooth
\cite{mar}.

In view of the above two-sided inequalities \eqref{jrho} and \eqref{krho}
and the lower bound \eqref{kjineq} it is natural to ask for which classes of
domains there is a majorant for $k_G$ in terms of $j_G.$
 Indeed, inequalities between
metrics can be used to define some important classes of domains. Thus, for instance, the class of simply connected plane domains $G$ for which there exists a constant $A \ge 1$ such that for all $x,y \in G$
\begin{equation} \label{unifdef}
k_G(x,y) \le A \, j_G(x,y)
\end{equation}
defines the class of quasidisks studied in \cite{gh}. Domains of this class, in turn, are
special cases of the so called $\varphi$-uniform domains \cite{hkv}.

\begin{nonsec}{\bf Dijkstra's algorithm compared to other path finding methods.}
Despite its age and simplicity, Dijkstra's algorithm remains one of the fastest and most used algorithms of computing distances in weighted graphs. If negative edge weights are allowed (but there are no negative cycles), the distances can be computed using the Bellman-Ford-Moore algorithm \cite{Bellman58,Ford56,Moore59}. If the digraph is unweighted, it is faster to use a simple algorithm adapted from breadth-first-search. During the preparation of this article, Duan et al. \cite{Duan25} showed that there exists an algorithm running in $\mathcal{O} (m \log^{2/3} n)$ time for computing the distances from one vertex to all others in a weighted digraph with non-negative edge weights. This is the long-awaited improvement to Dijkstra's algorithm, which has turned out difficult to improve due to the sorting step. However, Duan et al. make several assumptions to simplify the presentation of their algorithm. Most notably, the graphs are assumed to be sparse and every vertex has constant in- and out-degree. The algorithm they present is generalizable to graphs without these constraints. Dijkstra's algorithm is readily available and well-optimized for different programming languages, which is why we chose to use Dijkstra's algorithm and not implement Duan et al.'s algorithm for the purposes of our experiments.
\end{nonsec}

We discretize the problem by forming a grid of points in the domain $G$ (``fine enough'' with respect to the locations of points and the geometric features of $G$). Using these grid points, we define a graph and define the distance between two neighbouring points ``in a manner that is compatible with the quasihyperbolic geometry.'' The main step is then to find the shortest path in this graph; for this we use Dijkstra's algorithm. This yields approximations of quasihyperbolic geodesics by polygonal curves (shortest graph paths), and corresponding approximate quasihyperbolic distances, which are then compared to approximations obtained by other methods.

Question:
Let $G\subsetneq \mathbb{R}^n$ with $n\ge 2$, and for $x,y\in G$ define the quasihyperbolic distance by
\[
k_G(x,y)=\inf_{\gamma}\int_\gamma \frac{|dz|}{d(z,\partial G)},
\]
where the infimum is taken over all rectifiable curves $\gamma\subset G$ joining $x$ and $y$. Since the explicit value of the quasihyperbolic distance is available only in a few special cases, which existence statement holds in this setting?

There exist many comparison metrics, introduced over the past five decades, that are easy to evaluate and that provide either upper bounds or lower bounds for the quasihyperbolic metric.

Choices:
B. There exists a single comparison metric, introduced in the recent literature, that is easy to evaluate and that simultaneously gives both an upper bound and a lower bound for the quasihyperbolic metric in every proper domain.
C. There exist comparison metrics that are easy to evaluate and that provide either upper bounds or lower bounds for the quasihyperbolic metric.
D. There exist many comparison metrics, introduced over the past five decades, that are easy to evaluate and that provide both upper bounds and lower bounds for the quasihyperbolic metric.
E. There exist many comparison metrics, introduced over the past five decades, such that for every pair of points in every proper domain one can algorithmically compute the explicit value of the quasihyperbolic distance from these metrics.


\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)


\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 \(n\ge 2\), \(k\in\{1,2,\ldots,n-1\}\), \(L\ge 1\), and let \(K\subset\mathbb{R}^n\) be compact. For \(x\in K\), define the value function
\[
v(x):=\sup_{\mathrm P\in\mathcal P_x}\ \mathrm P\text{-ess}\inf\,\tau_K,
\]
where \(\tau_K\) is the first exit time of the coordinate process from \(K\), and \(\mathcal P_x\) is the admissible family of martingale laws starting from \(x\). Define, for \(p\in\mathbb R^n\) and \(M\in\mathbb S^n\),
\[
F(p,M):=\inf\Big\{-\frac12\operatorname{tr}(Ma):\ a\succeq 0,\ ap=0,\ \lambda_{(n-k)}(a)\ge 1,\ \lambda_{(1)}(a)\le L\Big\},
\]
where \(a\succeq 0\) means that \(a\) is positive semidefinite and \(\lambda_{(j)}(a)\) denotes the \(j\)-th largest eigenvalue of \(a\).

For a bounded function \(u:K\to\mathbb R\), let \(u^*\) and \(u_*\) denote its upper and lower semicontinuous envelopes, and let \(F_*\) and \(F^*\) denote the lower and upper semicontinuous envelopes of \(F\). Say that \(u\) is a viscosity subsolution of \(F(\nabla u,\nabla^2u)=1\) in \(K^\circ\) if, whenever \(x\in K^\circ\) and \(\varphi\in C^2(\mathbb R^n)\) satisfy
\[
(u^*-\varphi)(x)=\max_K (u^*-\varphi),
\]
one has
\[
F_*(\nabla\varphi(x),\nabla^2\varphi(x))\le 1.
\]
It satisfies the zero boundary condition on the subsolution side if the same implication holds for \(x\in\partial K\) whenever \(u^*(x)>0\). Define viscosity supersolution analogously by requiring that, whenever \(x\in K^\circ\) and \(\varphi\in C^2(\mathbb R^n)\) satisfy
\[
(u_*-\varphi)(x)=\min_K (u_*-\varphi),
\]
one has
\[
F^*(\nabla\varphi(x),\nabla^2\varphi(x))\ge 1,
\]
and it satisfies the zero boundary condition on the supersolution side if the same implication holds for \(x\in\partial K\) whenever \(u_*(x)<0\). A viscosity solution with zero boundary condition is a bounded function that is both such a subsolution and such a supersolution.

Suppose in addition that there exist maps \(T_\iota:\mathbb R^n\to\mathbb R^n\), \(\iota\in(1,2]\), each given by a composition of a rotation, a dilation, and a translation, such that \(K\subset \operatorname{int}(T_\iota(K))\) and \(\lim_{\iota\downarrow 1}T_\iota=I\), where \(I\) is the identity map on \(\mathbb R^n\). Which of the following conclusions about \(v\) holds?

The value function \(v\) is upper semicontinuous and is a viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K\) with zero boundary condition in the stated sense. Moreover, under the additional assumption on the maps \(T_\iota\), there is exactly one upper semicontinuous viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K\) with zero boundary condition.

Choices:
B. The value function \(v\) is upper semicontinuous and is a viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K^\circ\) with zero boundary condition in the stated sense. Moreover, under the additional assumption on the maps \(T_\iota\), there is exactly one bounded viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K\) with zero boundary condition.
C. The value function \(v\) is upper semicontinuous and is a viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K\) with zero boundary condition in the stated sense.
D. The value function \(v\) is upper semicontinuous and is a viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K\) with zero boundary condition in the stated sense. Moreover, under the additional assumption on the maps \(T_\iota\), every upper semicontinuous viscosity subsolution of \(F(\nabla u,\nabla^2 u)=1\) on \(K\) with zero boundary condition coincides with every lower semicontinuous viscosity supersolution on \(K\).
E. The value function \(v\) is upper semicontinuous and is a viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K\) with zero boundary condition in the stated sense. Moreover, under the additional assumption on the maps \(T_\iota\), there is exactly one lower semicontinuous viscosity solution of \(F(\nabla v,\nabla^2 v)=1\) on \(K\) with zero boundary condition.

uniqueness class changed from upper semicontinuous to bounded viscosity solutions

dropped the additional uniqueness conclusion under the maps \(T_\iota\)

comparison conclusion strengthened from \(u\le w^*\) to equality of arbitrary sub- and supersolutions

semicontinuity side in the uniqueness statement changed from upper semicontinuous to lower semicontinuous