  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.
