Width of spherical convex bodies.

For every hemisphere K supporting a convex body C on the sphere S 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 S and regular spherical odd-gons of thickness at most $${\frac{\pi}{2}}$$ on S are reduced. We prove that every reduced smooth spherical convex body is of constant width. [ABSTRACT FROM AUTHOR]