Conditions for Lipschitzness of boundary normal vector, almost everywhere

Question

Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition

(Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique point in $c(x) \in C$ such that $\|x-c(x)\| = \mbox{dist}(x,C) := \inf_{c \in C}\|x-c\|$.

For example, see Theorem 1 of http://tzamfirescu.tricube.de/TZamfirescu-110.pdf.

In particular, if $C$ is a closed convex subset of $\mathbb R^n$, then it is a folklore fact that $C$ verifies he unique CPP everywhere.

For any $\epsilon > 0$, let $C^\epsilon := \{x \in \mathbb R^n \mid d(x,A) \le \epsilon\}$ be the $\epsilon$-expansion of $C$.

Question. Under what minimalistic conditions on $C$ does there exist a function $\epsilon:(0,\infty) \to [0,\infty)$, such that $\limsup_{L \to \infty} \epsilon(L) = 0$ and the mapping $u_C:x \mapsto (x-c(x))/\|x-c(x)\|$ is $L$-Lipschitz a.e on $\mathbb R^n \setminus C^{\epsilon(L)}$ for any sufficiently large $L>0$ ?

• I'm particularly interested in the case of sets of the form $C := \cap_{i \in I} C_i$ with $C_i := \{x \in \mathbb R^n \mid f_i(x) \le 0\}$, where $f_i:\mathbb R^n \to \mathbb R$ are sufficiently smooth functions. Even the case where $I$ is a singleton $C$ is the sublevel set of a sufficiently smooth function is already interesting.

• The case where $C$ is a closed convex subset of $\mathbb R^n$ has been solved here https://mathoverflow.net/a/412676/78539: it suffices to take $\varepsilon(L) = 2/L$.

Answer 1

Given a nonempty closed set $C\subseteq\mathbb{R}^n$, let $S_C\subseteq\mathbb{R}^n$ denote the set of points for which the closest point in $C$ is not unique. Suppose $C$ satisfies each of the following:

• $S_C$ is nonempty (i.e., $C$ is not convex),
• there exists $\rho>0$ such that $C^\rho\cap S_C=\emptyset$, and
• the closest point mapping $c\colon(\mathbb{R}^n\setminus S_C)\to C$ is continuous.

(Every nonempty closed nonconvex subset $C\subseteq\mathbb{R}^n$ with smooth boundary that I can think of satisfies these property.)

Let $r_C<\infty$ denote the supremum of such $\rho$. Below, we show that for every $\epsilon\in(0,r_C)$, it holds that $u_C$ is not Lipschitz on $\mathbb{R}^n\setminus (C^\epsilon\cup Z)$ for any null set $Z$.

Select $p\in\mathbb{R}^n$ of distance $r_C$ from $C$ for which there exist $x,y\in C$ with $x\neq y$ such that $$ \|x-p\|=\|y-p\|=r. $$ Then for each $t\in(0,1)$, the points $x(t):=tx+(1-t)p$ and $y(t):=ty+(1-t)p$ are of distance less than $r_C$ from $C$, and so $c(x(t))=x$ and $c(y(t))=y$. Furthermore, $$ u_C(x(t))=\frac{x(t)-c(x(t))}{\|x(t)-c(x(t))\|}=\frac{x(t)-x}{\|x(t)-x\|}=\frac{p-x}{\|p-x\|}, $$ and similarly $$ u_C(y(t))=\frac{p-y}{\|p-y\|}. $$ Notably, for small $t$, $x(t)$ and $y(t)$ are arbitrarily close to each other, while $u_C(x(t))$ and $u_C(y(t))$ remain a fixed distance apart.

Given $\epsilon\in(0,r_C)$, a null set $Z$, and a constant $L>0$, we now show that $u_C$ is not $L$-Lipschitz on $\mathbb{R}^n\setminus (C^\epsilon\cup Z)$. Fix $\delta>0$ to be selected later. Select $t_0\in(0,1/2)$ small enough so that $x(t_0)$ and $y(t_0)$ avoid $C^\epsilon$ and $\|x(t_0)-y(t_0)\|<\delta$. Select a continuous path $q\colon[0,1]\to \mathbb{R}^n\setminus S_C$ such that $q(0)=x(t_0)$, $q(1)=y(t_0)$, and $q(s)\not\in Z$ for almost every $s\in(0,1)$. By the continuity of $c$, there exist $s_0,s_1\in[0,1]$ such that $q(s_0),q(s_1)\not\in Z$ and $$ \max\Big\{\|q(s_0)-x(t_0)\|,\|q(s_1)-y(t_0)\|,\|c(q(s_0))-x\|,\|c(q(s_1))-y\|\Big\}<\delta. $$ As we will see, $q(s_0)$ and $q(s_1)$ witness that $u_C$ is not $L$-Lipschitz on $\mathbb{R}^n\setminus (C^\epsilon\cup Z)$.

After some straightforward manipulations, we have \begin{align*} \|u_C(q(s_0))-u_C(x(t_0))\| &=\bigg\|\frac{q(s_0)-c(q(s_0))}{\|q(s_0)-c(q(s_0))\|}-\frac{x(t_0)-x}{\|x(t_0)-x\|}\bigg\|\\ &\leq\frac{4\delta}{\|x(t_0)-x\|}=\frac{4\delta}{(1-t_0)r_C}\leq\frac{8\delta}{r_C}, \end{align*} and similarly $$ \|u_C(q(s_1))-u_C(y(t_0))\|\leq\frac{8\delta}{r_C}. $$ Then $$ \|u_C(q(s_0))-u_C(q(s_1))\| \geq\|u_C(x(t_0))-u_C(y(t_0))\|-\frac{16\delta}{r_C} =\frac{\|x-y\|-16\delta}{r_C}, $$ but $$ \|q(s_0)-q(s_1)\| \leq\|x(t_0)-y(t_0)\|+2\delta \leq 3\delta. $$ Selecting $\delta<\min\{\frac{1}{32},\frac{1}{6Lr_c}\}\cdot\|x-y\|$ then gives $$ \|u_C(q(s_0))-u_C(q(s_1))\|>L\|q(s_0)-q(s_1)\|, $$ as claimed.