Distance to a closed set. Is this result known?

Question

Given a closed set $\varnothing\neq E\subset\mathbb{R}^n$, let $\operatorname{Unp}(E)$ be the set if points $x\in\mathbb{R}^n$ for which there is a unique point $y\in E$ nearest to $x$. Clearly $E\subset\operatorname{Unp}(E)$. On the other hand, it is known that the Lebesgue measure of the set $\mathbb{R}^n\setminus\operatorname{Unp}(E)$ equals zero, $\lvert\mathbb{R}^n\setminus\operatorname{Unp}(E)\rvert=0$ (cf. Set of points with a unique closest point in a compact set).

Erdős [1] proved a much stronger result:

The set $\mathbb{R}^n\setminus\operatorname{Unp}(E)$ is contained in the sum of countably many surfaces of finite $(n-1)$-dimensional measure.

This is a beautiful and I think, not very well known theorem. However, from the results existing in the literature one can conclude:

Theorem. For any closed set $E\subset\mathbb{R}^n$ there are countably many $C^2$-graphs $\{ G_j\}_{j=1}^\infty$ such that $$ \mathcal{H}^{n-1}\Bigl((\mathbb{R}^n\setminus\operatorname{Unp}(E))\setminus\bigcup_{j=1}^\infty G_j\Bigr)=0. $$

Here $\mathcal{H}^{n-1}$ stands for the Hausdorff measure and by a $C^2$-graph I mean the graph of a $C^2$ function $f:\mathbb{R}^{n-1}\to\mathbb{R}$. $$ \{x\in\mathbb{R}^n\mathrel: x_i=f(x_1,\dotsc,x_{i-1},x_{i+1},\dotsc,x_n)\}. $$

While the theorem can be concluded from what is in the literature, I was not able to find a straightforward reference to this statement. I think this result is of considerable interest, but most of people who would find it interesting would have difficulty to find it in the literature.

Question. Do you know if this result (or something very similar) has been published anywhere?

[1] P. Erdős, On the Hausdorff dimension of some sets in Euclidean space. Bull. Amer. Math. Soc. 52 (1946), 107-109.

Answer 1

I used such results in my thesis on the topology of singularities arising in various optimal transport programs (Sections 4.2-3). Especially useful was the following article of G. Alberti:

Alberti, Giovanni. "On the structure of singular sets of convex functions." Calculus of Variations and Partial Differential Equations 2.1 (1994): 17-27.

Alberti's Theorem: If $f:\mathbb{R}^n \to \mathbb{R}$ is proper lower semicontinuous convex function, then the subsets $$S^k(f):=\{x\in \mathbb{R}^n ~|~\dim_{\mathscr{H}} \partial_{\bullet} f \geq k\}$$ can be covered by countably many $(n-k)$-dimensional DC-manifolds.

Here $\partial_{\bullet} f $ is the local subdifferential of $f$.

The assumption that the function $f$ is convex is not as restrictive as it appears, for we can always replace $f$ with a locally semiconvex function. See Proposition 4.3.4. in my thesis for details. And if the closed set $E$ is locally compact (of course it is), then $dist_E$ becomes locally semiconvex, and we can apply Alberti's theorem to conclude that indeed the medial axis (or $\mathbb{R}^n-Unp(E)$ in the OP's notation) has Hausdorff dimension $\leq n-1$ and can be covered by countably many $(n-1)$-dimensional $DC$-manifolds.

Related results occur in R.J.McCann's and J. Kitagawa's article "Free Discontinuities in Optimal Transport". https://arxiv.org/abs/1708.04152

Answer 2

The following result can be found in

D. H. Fremlin, Skeletons and central sets, Proc. London Math. Soc. (3) 74 (1997), 701–720.

Let $\Omega$ be an open proper subset of $\mathbb{R}^n$, where $n > 1$. Its skeleton is

$$R:=\{ x \in \Omega: \text{ there are distinct } y, y'\in \mathbb{R}^n \setminus \Omega \text{ such that } \rho( x , y ) =\rho( x , y ) = \rho( x , \mathbb{R}^n \setminus \Omega)\},$$ where we write $\rho$ for the Euclidean metric on $\mathbb{R}^n$.

Theorem 1G: Let $\Omega$ be an open proper subset of $\mathbb{R}^n$. Then its skeleton $R$ is the union of a sequence of closed sets each of which is Lipschitz isomorphic to a subset of $\mathbb{R}^{n-1}$. Consequently , the dimension of $R$ is at most $n-1$, whether we mean inductive dimension , covering dimension , Hausdorff dimension or Minkowski dimension.