Region of maximum probability content of fixed diameter (Q911139)

Let \(C_ d\) be the class of all convex compacta in \({\mathbb{R}}^ k\) which have diameter d, and let X be a random vector with probability density function with respect to the Lebesgue measure which is nonincreasing in \(| x_ i|\), \(i=1,...,k\). Denote the probability that X belongs to A by P(A). The maximum of P(A) for A of diameter d is to be found in \(C_ d\). The author proves this maximum is attained for the ball of radius d/2 centered at the origin and goes on to show how Bieberbach's inequality connecting the volume and diameter of a convex body follows from this result.