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Diameter of reduced spherical convex bodies

Lassak M., Musielak M.

Fasciculi Mathematici

2018

Nr 61

103--108

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) < π/2, in terms of its thickness.

Reduced spherical convex bodies

Lassak M., Musielak M.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 1

87--97

The paper presents a number of properties of reduced bodies on the two-dimensional sphere. Our main theorem describes the shape of reduced bodies of thickness below π/2.