Maximal distance minimizers for a rectangle

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