On some mean values associated with a randomly selected simplex in a convex set

Abstract

For any convex body K in euclidean %-space denote by m(K) the mean value of the volume of a simplex with vertices at n + 1 randomly selected points from K. It is shown that among all convex bodies of given volume the mean value m(K) is minimal if and only if K is an ellipsoid. Actually, a more general result is obtained which shows that the higher order moments of the volume of a randomly selected simplex in a convex set have similar minimal properties.