This is followup to the following question: On the Lipschitz continuity of the unit-normal vector field of a polytope

Let $C$ be a (nonempty) closed convex subset of $\mathbb R^n$. Note that to every $x \in \mathbb R^n$ corresponds a point $c(x) \in C$ which is closest to $x$. Let $C' := \mathbb R^n \setminus C$ and define $u:C' \to \mathbb R^n$ by $u(x) := (x-c(x))/\|x-c(x)\|$.

Question. What are generic conditions on $C$ under which $u_C$ is Lipschitz-continuous on $C'$ ?

Are there any such conditions linked with a "classical" notion of smoothness or curvature of (the boundary of) a closed convex set ?

Examples

• If $C := \{0\}$, then $u$ is the function $x \mapsto x/\|x\|$, which is definitely not Lipschitz on its domain $C'=\mathbb R^n\setminus\{0\}$.
• If $C = B_n$, the unit-ball in $\mathbb R^n$, then $u$ is the map $x \mapsto x/\|x\|$, which is $1$-Lipschitz on $C'$.
• If $A$ is a positive-definite matrix and $C = \{x \in \mathbb R^n \mid x^\top A x \le 1\}$, then $u$ is the map $x \mapsto Ax/\|Ax\|$, which is $\mbox{cond}(A)$-Lipschitz on $C'$, where $\mbox{cond}(A) := \|A\|_{op}\|A^{-1}\|_{op}$ is the condition number of $A$. The previous example is a particular case with $\mbox{cond}(A) = 1$.