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An overview of maximal distance minimizers problem

Danila Cherkashin, Yana Teplitskaya

Abstract

Consider a compact $M \subset \mathbb{R}^d$ 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 \[ \max_{y \in M} dist (y, Σ), \] 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.

An overview of maximal distance minimizers problem

Abstract

Consider a compact and . A maximal distance minimizer problem is to find a connected compact set of the length (one-dimensional Hausdorff measure ) at most that minimizes where 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 -convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.
Paper Structure (39 sections, 23 theorems, 30 equations, 8 figures)

This paper contains 39 sections, 23 theorems, 30 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Class of problems
  3. Dual problem
  4. The first parallels with the average distance minimization problem
  5. Notation
  6. Existence. Absence of loops. Ahlfors regularity and other simple properties
  7. Complexity of the problem
  8. Local maximal distance minimizers
  9. Regularity
  10. Tangent rays. Blow up limits in Rd
  11. Properties of branching points in R2
  12. Continuity of one-sided tangents in R2
  13. Planar example of infinite number of corner points
  14. Every C11-smooth simple curve is a minimizer
  15. Explicit examples for maximal distance minimizers
  16. ...and 24 more sections

Key Result

Theorem 2.2

Let $\Sigma$ be a minimizer for a compact set $M \subset \mathbb{R}^d$ and $r > 0$. Then there are at most three tangent rays at any point of $\Sigma$, and the pairwise angles between the tangent rays are at least $2\pi/3$. Furthermore, tangent rays coincide with one-sided tangents, particularly the

Figures (8)

  • Figure 1: All possible variants of tangent rays at any point of a maximal distance minimizer or blow up limits of an average distance minimizer
  • Figure 2: The example of a minimizer with infinite number of corner points
  • Figure 3: A maximal distance minimizer for a certain $3$-point set $M = \{a,b,c\}$
  • Figure 4: An example to Proposition \ref{['prop:notaminimizer']}
  • Figure 5: A minimizer for a convex closed planar curve $M$ with the radius of curvature at least $5r$ at every point, so-called horseshoe (left). A minimizer for $M = \partial B_R(x)$, where $R > 4.98r$ (right)
  • ...and 3 more figures

Theorems & Definitions (36)

  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Theorem 2.2: Gordeev--Teplitskaya gordeev2022regularity
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6: Lemenant, 2011 lemenant2011regularity
  • Corollary 2.7
  • Theorem 2.8: Basok--Cherkashin--Teplitskaya, 2022 inverse2022
  • ...and 26 more