Reduced spherical polygons

Marek Lassak¹

¹ Institute of Mathematics and Physics University of Science and Technology 85-789 Bydgoszcz, Poland

Colloquium Mathematicum, Tome 138 (2015) no. 2, pp. 205-216

Résumé

For every hemisphere $K$ supporting a spherically convex body $C$ of the $d$-dimensional sphere $S^d$ we consider the width of $C$ determined by $K$. By the thickness $\varDelta (C)$ of $C$ we mean the minimum of the widths of $C$ over all supporting hemispheres $K$ of $C$. A spherically convex body $R \subset S^d$ is said to be reduced provided $\varDelta (Z) \varDelta (R)$ for every spherically convex body $Z \subset R$ different from $R$. We characterize reduced spherical polygons on $S^2$. We show that every reduced spherical polygon is of thickness at most $\pi /2$. We also estimate the diameter of reduced spherical polygons in terms of their thickness. Moreover, a few other properties of reduced spherical polygons are given.

Détail

DOI : 10.4064/cm138-2-5

Keywords: every hemisphere supporting spherically convex body d dimensional sphere consider width determined thickness vardelta mean minimum widths supporting hemispheres spherically convex body subset said reduced provided vardelta vardelta every spherically convex body subset different characterize reduced spherical polygons every reduced spherical polygon thickness estimate diameter reduced spherical polygons terms their thickness moreover few other properties reduced spherical polygons given