Inequalities for probability contents of convex sets via geometric average

It is shown that: If (X1, X2) is a permutation invariant central convex unimodal random vector and if A is a symmetric (about 0) permutation invariant convex set then P{(aX1, X2/a) [set membership, variant] A} is nondecreasing as a varies from )+ to 1 and is non-increasing as a varies from 1 to [infinity] (that is, P{(a1X1, a2X2) [epsilon] A} is a Schur-concave function of (log a1, log a2). Some extensions of this result for the n-dimensional case are discussed. Applications are given for elliptically contoured distributions and scale parameter families.