Maximal distance minimizers for a rectangle

\emph{A maximal distance minimizer} for a given compact set $M \subset $\mathbb{R}$^2$andsomegiven$r > 0$ is a set having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\mathrm{\Sigma }\subset {\mathbb{R}}^2$ satisfying the inequality $$\underset{y\in M}{\max }dist(y,\mathrm{\Sigma })\le r\mathrm{.}$$ This paper deals with the set of maximal distance minimizers for a rectangle $M$ and small enough $r$.