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- W4299492246 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." @default.
- W4299492246 created "2022-10-02" @default.
- W4299492246 creator A5000434309 @default.
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- W4299492246 date "2015-11-03" @default.
- W4299492246 modified "2023-09-27" @default.
- W4299492246 title "On the horseshoe conjecture for maximal distance minimizers" @default.
- W4299492246 doi "https://doi.org/10.48550/arxiv.1511.01026" @default.
- W4299492246 hasPublicationYear "2015" @default.
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