The Maximum Distance Problem and Minimal Spanning Trees

Given a compact $E\subset \mathbb{R}^n$ and $s > 0$, the maximum distance problem asks to find 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 minimal 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 in $\mathbb{R}^2$, with a finite number balls of radius $s$, and connect their centers with another Lipschitz curve, $\Gamma_\ast$, where $\mathcal{H}^1(\Gamma_\ast)$ is arbitrarily close to $\mathcal{H}^1(\Gamma)$.