Inequalities of the Brunn-Minkowski type for Gaussian measures

Abstract

Let m 2 be an integer, let γ be the standard Gaussian measure on Rn}, and let Φ(t)=∫-∞}t exp (-s2/2)ds sqrt{2π}{small} -∞ t Le ∞. Given α 1}l̇, αm} ] 0,∞ this paper gives a necessary and sufficient condition such that the inequality Φ-1} (γ (α1}A1}+ċ+αm}A m} α1}Φ-1}(γA 1)+ċ+ αm}Φ-1}(γA m) is true for all Borel sets A 1,...,A m in hbfRn} of strictly positive γ-measure or all convex Borel sets A 1,...,A m in bfRn} of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn-Minkowski type for γ which are true for all convex sets but not for all measurable sets. © 2007 Springer-Verlag.