2016/07/01 by Marek Lassak, Lassak, Marek, Michał Musielak +1 · 1 citation
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #Primary 52A55 #Secondary 97G60 #math.MG #msc:52A55 #msc:97G60
paper · pdf · doi:10.48550/arxiv.1607.00132
arxiv created 2016/07/01 · arxiv updated 2016/07/04
The aim of this paper is to present some properties of reduced spherical convex bodies on the two-dimensional sphere S2. The intersection of two different non-opposite hemispheres is called a lune. By its thickness we mean the distance of the centers of the two semicircles bounding it. The thickness Δ(C) of C is the minimum thickness of a lune containing C. We say that a spherical convex body R is reduced if Δ(Z) < Δ(R) for every spherical convex body Z ⊂ R different from R. Our main theorem permits to describe the shape of reduced bodies of thickness below \fracπ2. It implies a number of corollaries. In particular, we estimate the diameter of reduced spherical bodies in terms of their thickness. Reduced bodies of thickness at least \fracπ2 have constant width. Spherical convex bodies of constant width below \fracπ2 are strictly convex.