Published Paper

Inserted: 10 feb 2023
Last Updated: 22 oct 2025

Journal: Serdica Journal of Computing
Volume: 18
Number: 1
Pages: 125-155
Year: 2024
Doi: 10.55630/sjc.2024.18.125-155

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 $\H$) 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.

Keywords: maximal distance minimizer