On minimizers of the maximal distance functional for a planar convex closed smooth curve

TL;DR

This paper studies the structure of minimal-length connected sets that cover a convex curve within a fixed distance, revealing local Steiner tree properties and behavior as points move along the curve.

Contribution

It characterizes the structure of maximal distance minimizers for convex curves, showing they relate to local Steiner trees with limited vertices and analyzing their behavior under point movement.

Findings

Closure of connected components forms local Steiner trees with up to five vertices.

Connected sets are structured around local Steiner trees.

Behavior of minimizers is analyzed as points move along the curve.

Abstract

Fix a compact $M \subset \mathbb{R}^2$ and $r>0$. A minimizer of the maximal distance functional is a connected set $\Sigma$ of the minimal length, such that \[ max_{y \in M} dist(y,\Sigma) \leq r. \] The problem of finding maximal distance minimizers is connected to the Steiner tree problem. In this paper we consider the case of a convex closed curve $M$, with the minimal radius of curvature greater than $r$ (it implies that $M$ is smooth). The first part is devoted to statements on structure of $\Sigma$: we show that the closure of an arbitrary connected component of $B_r(M) \cap \Sigma$ is a local Steiner tree which connects no more than five vertices. In the second part we "derive in the picture". Assume that the left and right neighborhoods of $y \in M$ are contained in $r$-neighborhoods of different points $x_1$, $x_2 \in \Sigma$. We write conditions on the behavior of $\Sigma$ in the neighborhoods of $x_1$ and $x_2$ under the assumption by moving $y$ along $M$.