On the volume of convex hulls of sets on spheres

We prove that for a measurable subset of S^n–1 with fixed Haar measure, the volume of its convex hull is minimized for a cap (i.e. a ball with respect to the geodesic measure). We solve a similar problem for symmetric sets and n=2, 3. As a consequence, we deduce a result concerning Gaussian measures of dilatations of convex, symmetric sets in R² and R³.Partially supported by KBN (Poland), Grant No. 2 1094 91 01.