Maximal distance minimizers for a rectangle

TL;DR

This paper investigates the properties and structure of minimal-length connected sets that ensure every point in a rectangle is within a certain distance, focusing on the case of small radius values.

Contribution

It characterizes the maximal distance minimizers specifically for rectangles when the radius is sufficiently small.

Findings

Identifies the structure of minimal sets for rectangles

Provides conditions for small radius values

Advances understanding of distance minimization problems

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