TITLE: Distance to a closed set. Is this result known?
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.