An overview of maximal distance minimizers problem

TL;DR

This paper surveys the maximal distance minimizers problem, discussing its properties, computational complexity, convergence, and open questions in the context of geometric optimization.

Contribution

It provides a comprehensive overview, proves NP-hardness, establishes $ ext{Gamma}$-convergence, and explores solution uniqueness for the problem.

Findings

NP-hardness of the maximal distance minimizers problem

Establishment of $ ext{Gamma}$-convergence results

Discussion of solution uniqueness and open questions

Abstract

Consider a compact $M \subset \mathbb{R}^d$ and $l > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the length (one-dimensional Hausdorff measure $\mathcal H$) at most $l$ that minimizes \[ \max_{y \in M} dist (y, \Sigma), \] 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 $\Gamma$-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.