On Convexity of Measures

Abstract

A simple geometric proof and some applications are given to results of C. Borell providing necessary and sufficient conditions that a density in $R^n$ generates a measure satisfying a convexity property of the type $$P(\theta A_0 + (1 - \theta)A_1) \geqq \{\theta\lbrack P(A_0)\rbrack^s + (1 - \theta)\lbrack P(A_1) \rbrack^s\}^{1/s}.$$