Let $\mu$ be a probability measure on $\mathbb R^n$ and let $P$ be a compact polytope 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 for every $\epsilon>0$, there exists a measurable set $P^\epsilon$ such that $P \subseteq P^\epsilon$, $\mu(P^\epsilon) \le \mu(P) + \epsilon$, and $x \mapsto u(x)$ is Lipschiitz-continuous.

For example, if $P$ is just a half-space, $u$ is a constant, and thus Lipschitz-continuous on $\mathbb R^n\setminus P$.

Related. Conditions for Lipschitzness of boundary normal vector, almost everywhere