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On the horseshoe conjecture for maximal distance minimizers
We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ⊂R2satisfying the inequality maxy∈Mdist(y,Σ)≤r for a given compact set M⊂R2 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 Σ has similar structure for r<R/5. Additionaly we prove a similar statement for local minimizers.