Inequalities for the Gaussian measure of convex sets

Michael R Tehranchi

doi:10.1214/17-ecp89

Abstract

This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.

Highlights

Introduction and statement of resultsLet γ be the standard Gaussian on Rn, defined by γ(K) = 1 (2π)n/2 e− 1 2 x dxK for Lebesgue measurable K ⊆ Rn
We present the proof
In Appendix A we provide an interesting reformulation of this monotonicity property in terms of the function sinc x

Summary

The proof

For 0 ≤ t ≤ 1 let γt denote the distribution on Rn × Rn of the jointly normal vector (X, Y ) where the distribution of both X and Y is the standard Gaussian measure γ and the covariance matrix is E(XY ) = tI where I is the n × n identity matrix. This family of measures (γt)0≤t≤1 interpolates between γ0(K × L) = γ(K)γ(L) and γ1(K × L) = γ(K ∩ L) and is given explicitly, for t < 1, by the formula γt(H) =. Γ(A)γ(B) ≤ (1 − S)−n/2γt 1 − S(A × B)

A A sinc reformulation

B Log-concave functions