A nonsymmetric correlation inequality for Gaussian measure

Let mu be a Gaussian measure (say, on R-n) and let K,L subset of or equal t o R-n be such that K is convex, L is a "layer" (i.e., L = {x: a less than o r equal to < x, u > less than or equal to b} for some a, b is an element of <(R) over bar> and u is an element of R-n) and the centers of mass (with r espect to mu) of K and L coincide. Then mu(K boolean AND L) greater than or equal to mu(K) . mu(L). This is motivated by the well-known "positive corr elation conjecture" for symmetric sets and a related inequality of Sidak co ncerning confidence regions for means of multivariate normal distributions. The proof uses the estimate Phi(x) > 1 - ((8/pi)(1/2)/(3x + (x(2) + 8)(1/2 )))e-(x2/2), x > -1, for the (standard) Gaussian cumulative distribution fu nction. which is sharper than the classical inequality of Komatsu. (C) 1999 Academic Press AIMS 1991 subject classifications: 62H20, 52A20, 46B09, 60E 15, 60D05.