The Maximum Distance Problem and Minimal Spanning Trees

arXiv2004.07323MaRDI QIDQ6338781

Enrique G. Alvarado, Bala Krishnamoorthy, Kevin R. Vixie

Publication date: 15 April 2020

Abstract: Given a compact $Esubsetmathbb{R}^n$ 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 $Esubsetmathbb{R}^2$, 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 $Gamm{a}_{a}st$, where $mathcal{H}^1(Gamm{a}_{a}st)$ 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.

Has companion code repository: https://github.com/mtdaydream/MDP_MST

Mathematics Subject Classification ID

Variational problems in a geometric measure-theoretic setting (49Q20) Geometric measure and integration theory, integral and normal currents in optimization (49Q15) Length, area, volume, other geometric measure theory (28A75)

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