The Maximum Distance Problem and Minimal Spanning Trees

Abstract

Given a compact $E\subset {\mathbb{R}}^n$ and $s>0$, the maximum distance problem seeks a compact and connected subset of ${\mathbb{R}}^n$ of smallest one dimensional Hausdorff measure whose $s$-neighborhood covers $E$. For $E\subset {\mathbb{R}}^2$, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius $s$, which cover $E$, solves the maximum distance problem. The main difficulty in proving this result is overcome by the proof of Lemma 3.5, which states that one is able to cover the $s$-neighborhood of a Lipschitz curve $\Gamma$ in ${\mathbb{R}}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve ${\Gamma }_{\ast }$, where ${\mathcal{H}}^1({\Gamma }_{\ast })$ is arbitrarily close to ${\mathcal{H}}^1(\Gamma )$. We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at https://github.com/mtdaydream/MDP_MST.

Remarks 1.5, 3.6 added to compare with result of Miranda Jr. et al. (2006)