Ehrhard’s inequality is a sharp inequality about the Gaussian measure of Minkowski sum of sets. We put forward a plausible conjecture in regards to the sharp version of the Ehrhard’s inequality when the sets are symmetric and convex. We explain that this conjecture follows from showing that certain functional of a convex set is minimized on “k-round cylinders” — direct products of balls and subspaces (which are also conjectural minimizers for the Gaussian perimeter of symmetric sets, as per works of Morgan and Heilman). We provide certain different lower estimate for this functional, and characterize the equality cases to be k-round cylinders. Along the way, we establish equality cases in the (general form of the) Brascamp-Leib inequality on convex sets. Methods combine variational calculus with L2 estimates.
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