Maximal distance minimizers for rectangle

Fix a compact $M \subset \mathbb R^2$ and $r>0$. Maximal distance minimizer is a connected set $\Sigma$ of the minimal length such that
$$ \max_{y \in \mathbb M} dist(y, \Sigma)\le r, $$
i.e. $M \subset B_r (\Sigma)$.
We determine the set of maximal distance minimizers for rectangle and small enough $r$.

Theorem. Let $M$ be a rectangle, $0<r<r_0(M)$. Then maximal distance minimizer is unique (up to symmetries of $M$). It is depicted on the picture "Theorem" (the right part of the picture contains enlarged fragment of the minimizer; the marked angle tends to $\frac {11\pi}{12}$ with $r \to \infty$).

Joint work with A. Gordeev, G. Strukov and Y. Teplitskaya.