Let $P$ be a closed polytope (i.e the intersection of a finite number of closed half-spaces) in $\mathbb R^n$. For any $x \in \mathbb R^n \setminus P$, let $p(x) \in P$ be (unique!) point in $P$ which is closest to $x$ and let $d(x) := \|x-p(x)\|$ be the distance from $x$ to $p$ and $u(x) := (x-p(x))/d(x) \in S_{n-1}$, where $S_{n-1}$ is the unit-sphere in $\mathbb R^n$.

>**Question.** *Is it true that that the mapping $x \mapsto u(x)$ is Lipschiitz continuous almost everywhere on $\mathbb R^n \setminus P$.*

For example, if $P$ is just a half-space, $u$ is a constant, and thus Lipschitz-continuous.

**Related.** https://mathoverflow.net/q/412656/78539