On minimizers of the maximal distance functional for a planar convex closed smooth curve

2020/11/20 by Cherkashin, D. D., Gordeev, A. S., Strukov, G. A. +1
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2011.10463

Abstract

Fix a compact M ⊂ ℝ² and r>0. A minimizer of the maximal distance functional is a connected set Σ of the minimal length, such that max_{y ∈ M} dist(y,Σ) ≤ r. The problem of finding maximal distance minimizers is connected to the Steiner tree problem. In this paper we consider the case of a convex closed curve M, with the minimal radius of curvature greater than r (it implies that M is smooth). The first part is devoted to statements on structure of Σ: we show that the closure of an arbitrary connected component of B_r(M) ∩ Σ is a local Steiner tree which connects no more than five vertices. In the second part we "derive in the picture". Assume that the left and right neighborhoods of y ∈ M are contained in r-neighborhoods of different points x₁, x₂ ∈ Σ. We write conditions on the behavior of Σ in the neighborhoods of x₁ and x₂ under the assumption by moving y along M.

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