Given a compact E \subset R^n and s > 0, the maximum distance problem asks to find a compact and connected subset of R^n of smallest one dimensional Hausdorff measure , whose s-neighborhood covers E. For E\subset 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 R^2 , with a finite number  balls of radius s, and connect their centers with another Lipschitz curve , \Gamma_\ast, where H^1(\Gamma_\ast) is arbitrarily close to H^1(\Gamma). 
Given a compact E \subset R^n and s > 0, the maximum distance problem seeks a compact and connected subset of R^n of smallest one dimensional Hausdorff measure whose s-neighborhood covers E. For E\subset 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 R^2 with a finite number of balls of radius s, and connect their centers with another Lipschitz curve \Gamma_\ast, where H^1(\Gamma_\ast) is arbitrarily close to H^1(\Gamma). We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at URL
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We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at URL