Reduced Spherical Convex Bodies

TL;DR

This paper investigates properties of reduced convex bodies on the sphere, characterizing their shape, diameter, and convexity features, especially for bodies with thickness below or above rac{a}{2}.

Contribution

It provides a detailed description of reduced spherical convex bodies of certain thicknesses and establishes their geometric properties and relationships.

Findings

Reduced bodies of thickness less than a/2 have a specific shape characterized by the main theorem.

The diameter of reduced bodies can be estimated based on their thickness.

Bodies with thickness at least a/2 have constant width, and those below a/2 are strictly convex.

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.