A Nonsymmetric Correlation Inequality for Gaussian Measure

Letμbe a Gaussian measure (say, onRn) and letK,L⊆Rnbe such thatKis convex,Lis a “layer” (i.e.,L={x: a⩽〈x, u〉⩽b} for somea, b∈Randu∈Rn), and the centers of mass (with respect toμ) ofKandLcoincide. Thenμ(K∩L)⩾μ(K)·μ(L). This is motivated by the well-known “positive correlation conjecture” for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimateΦ(x)> 1−((8/π)1/2/(3x+(x2+8)1/2))e−x2/2,x>−1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu.