Symmetric Gaussian Problem

Determine, for fixed 0 ≤ ρ, μ, ν ≤ 1 and centrally symmetric A ⊆ R^n of Gaussian measure μ and B ⊆ R^n of Gaussian measure ν, the minimal possible value of Pr[X ∈ A, Y ∈ B] when (X, Y) are ρ-correlated n-dimensional Gaussians.

Background

Without symmetry, Borell’s inequality shows that opposing halfspaces minimize the joint probability. With the symmetry constraint A = −A (equivalently both A and B symmetric), the extremal sets are unknown.

A natural candidate extremizer is the pair consisting of a centered ball and the complement of a centered ball, which would mirror certain isoperimetric behaviors under correlation.