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) A} is nondecreasing as a varies from )+ to 1 and is non-increasing as a varies from 1 to ∞ (that is, P{(a₁X₁, a₂X₂) ε 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.