On the horseshoe conjecture for maximal distance minimizers

Abstract

We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality maxy∈M dist (y,Σ) ≤ r for a given compact set M ⊂ ℝ2 and some given r > 0. Such sets play the role of shortest possible pipelines arriving at a distance at most r to every point of M, where M is the set of customers of the pipeline. We describe the set of minimizers for M a circumference of radius R > 0 for the case when r < R ∕ 4 .98, thus proving the conjecture of Miranda, Paolini and Stepanov for this particular case. Moreover we show that when M is the 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.