The Gaussian correlation conjecture states that for any two
symmetric, convex sets in $n$-dimensional space and for any centered,
Gaussian measure 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 standard
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 radius $c
\sqrt{n}$. Further we show that if for every $n$ the conjecture is true
when the sets are in the Euclidean ball of radius $\sqrt{n}$, then it is true
in general. Our most concrete result is that the conjecture is true if the two
sets are (arbitrary) centered ellipsoids.