The Gaussian correlation conjecture states that for any two symmetric,
convex sets in n-dimensional space and for any centered, Gaussian mea
sure on that space, the measure of the intersection is greater than or
equal to the product of the measures. In this paper we obtain several
results which substantiate this conjecture. For example, in the stand
ard Gaussian case, we show there is a positive constant, c, such that
the conjecture is true if the two sets are in the Euclidean ball of ra
dius c root n. Further we show that if for every n the conjecture is t
rue when the sets are in the Euclidean ball of radius root n, then it
is true in general. Our most concrete result is that the conjecture is
true if the two sets are (arbitrary) centered ellipsoids.