An overview of maximal distance minimizers problem

2022/12/11 by Cherkashin, Danila, Teplitskaya, Yana · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2212.05607

Abstract

Consider a compact M ⊂ ℝ^d and l > 0. A maximal distance minimizer problem is to find a connected compact set Σ of the length (one-dimensional Hausdorff measure \mathcal H) at most l that minimizes max_{y ∈ M} dist (y, Σ), where dist stands for the Euclidean distance. We give a survey on the results on the maximal distance minimizers and related problems. Also we fill some natural gaps by showing NP-hardness of the maximal distance minimizing problem, establishing its Γ-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.

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