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A Gaussian correlation inequality for certain bodies in ? n
If $n(x)$ is the standard normal density on $R^2$ and if $A = -A$ and $B = -B$ are convex subsets of $R^2$ then $$\int_{A\cap B}\mathbf{n}(x) d^2x \geqq (\int_A \mathbf{n}(x) d^2x)(\int_B \mathbf{n}(x) d^2x).$$
Gaussian correlation inequalities for ferromagnets