Inverse maximal and average distance minimizer problems

Consider a compact $M \subset \mathbb{R}^d$ and $r > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the minimal length, such that $$\max_{y \in M} dist (y, \Sigma) \leq r.$$ The inverse problem is to determine whether a given compact connected set $\Sigma$ is a minimizer for some compact $M$ and some positive $r$. Let a Steiner tree $St$ with $n$ terminals be unique for its terminal vertices. The first result of the paper is that $St$ is a minimizer for a set $M$ of $n$ points and a small enough positive $r$. It is known that in the planar case a general Steiner tree (on a finite number of terminals) is unique. It is worth noting that a Steiner tree on $n$ terminal vertices can be not a minimizer for any $n$ point set $M$ starting with $n = 4$; the simplest such example is a Steiner tree for the vertices of a square. It is known that a planar maximal distance minimizer is a finite union of simple curves. The second result is an example of a minimizer with an infinite number of corner points (points with two tangent rays which do not belong to the same line), which means that this minimizer can not be represented as a finite union of smooth curves. Our third result is that every injective $C^{1,1}$-curve $\Sigma$ is a minimizer for a small enough $r>0$ and $M = \overline{B_r(\Sigma)}$. The proof is based on analogues result by Tilli on average distance minimizers. Finally, we generalize Tilli's result from the plane to $d$-dimensional Euclidean space.