Calculus of Variations and Geometric Measure Theory

D. Cherkashin - Y. Teplitskaya

On the horseshoe conjecture for maximal distance minimizers

created by teplitskaya1 on 06 Jun 2021
modified by cherkashin on 12 Feb 2024



Inserted: 6 jun 2021
Last Updated: 12 feb 2024

Journal: ESAIM: COCV
Volume: 24
Number: 3
Pages: 1015–1041
Year: 2015

ArXiv: 1511.01026 PDF


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.

Keywords: explicit solution, maximal distance minimizer