Width of spherical convex bodies

For every hemisphere K supporting a convex body C on the sphere S d 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 Sᵈ equals the maximum of the widths of C provided the diameter of C is at most {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 Sᵈ 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 d and regular spherical odd-gons of thickness at most {2} on S 2 are reduced. We prove that every reduced smooth spherical convex body is of constant width.