Diameter of reduced spherical convex bodies

2018/11/06 by Lassak, Marek, Musielak, Michał
#52A55 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1811.02487

Abstract

The intersection L of two different non-opposite hemispheres of the unit sphere S² is called a lune. By Δ(L) we denote the distance of the centers of the semicircles bounding L. By the thickness Δ(C) of a convex body C ⊂ S² we mean the minimal value of Δ(L) over all lunes L ⊃ C. We call a convex body R⊂ S² reduced provided Δ(Z) < Δ(R) for every convex body Z being a proper subset of R. Our aim is to estimate the diameter of R, where Δ(R) < \fracπ2, in terms of its thickness.

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