Reduced Spherical Convex Bodies

Abstract

The aim of this paper is to present some properties of reduced spherical convex bodies on the two-dimensional sphere $S^2$. 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 $\Delta (C)$ of $C$ is the minimum thickness of a lune containing $C$. We say that a spherical convex body $R$ is reduced if $\Delta (Z) < \Delta (R)$ for every spherical convex body $Z \subset R$ different from $R$. Our main theorem permits to describe the shape of reduced bodies of thickness below $\frac{\pi}{2}$. 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 $\frac{\pi}{2}$ have constant width. Spherical convex bodies of constant width below $\frac{\pi}{2}$ are strictly convex.