REDUCED SPHERICAL POLYGONS

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 Delta(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 of S-d is said to be reduced provided Delta(Z) < Delta(R) for every spherically convex body Z subset of 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.