Maximal distance minimizers for a rectangle

Abstract

\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 \[ \max_{y\in M} dist (y, Σ) \leq r. \] This paper deals with the set of maximal distance minimizers for a rectangle $M$ and small enough $r$.

25p, 17f