On the horseshoe conjecture for maximal distance minimizers

TL;DR

This paper proves the horseshoe conjecture for minimal distance minimizers in the plane, showing the structure of optimal sets under certain geometric constraints and for specific shapes of the set M.

Contribution

It confirms the horseshoe conjecture for the case when M is a circle or convex boundary, establishing the structure of minimizers for small r values.

Findings

Proved the conjecture for M as a circle when r < R/4.98.

Established the structure of minimizers for convex boundaries when r < R/5.

Extended results to local minimizers with similar properties.

Abstract

We study the properties of sets $\Sigma$ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality $\mbox{max}_{y \in M} \mbox{dist}(y,\Sigma) \leq r$ for a given compact set $M \subset \mathbb{R}^2$ and some given $r > 0$. Such sets can be considered shortest possible pipelines arriving at a distance at most $r$ to every point of $M$ which in this case is considered as the set of customers of the pipeline. We prove the conjecture of Miranda, Paolini and Stepanov about the set of minimizers for $M$ a circumference of radius $R>0$ for the case when $r < R/4.98$. Moreover we show that when $M$ is a boundary of a smooth convex set with minimal radius of curvature $R$, then every minimizer $\Sigma$ has similar structure for $r < R/5$. Additionaly we prove a similar statement for local minimizers.