Width of spherical convex bodies

Abstract

For every hemisphere K supporting a convex body C on the sphere Sd we define the width of C determined by K. We show that it is a continuous function of the position of K. We prove that the diameter of every convex body \({C \subset S^d}\) equals the maximum of the widths of C provided the diameter of C is at most \({\frac{\pi}{2}}\). In a natural way, we define spherical bodies of constant width. We also consider the thickness Δ(C) of C, i.e., the minimum width of C. A convex body \({R \subset S^d}\) is said to be reduced if Δ(Z) < Δ(R) for every convex body Z properly contained in R. For instance, bodies of constant width on Sd and regular spherical odd-gons of thickness at most \({\frac{\pi}{2}}\) on S2 are reduced. We prove that every reduced smooth spherical convex body is of constant width.