On the horseshoe conjecture for maximal distance minimizers

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 $\mbox{max}_{y \in M} \mbox{dist}(y,Σ) \leq r$ 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.

25 pages, 21 figures