We shall discuss Brunn-Minkowski type inequalities for probability measures. We provide a sharp lower estimate for the so-called concavity exponent of a symmetric convex set, and show that round k-cylinders (direct products of balls and subspaces of lower dimension) are the only equality cases in this inequality. This relies on the equality case characterization in the Brascamp-Lieb inequality for restrictions of Gaussian measure to smooth convex sets, as well as on various aspects of Gaussian energy minimization. We further provide explicit non-sharp lower bounds for the exponent in the Gaussian Brunn-Minkowski inequality for symmetric convex sets whose measure is explicitly lower-bounded from below, with applications to isoperimetric-type inequalities. Lastly, we discuss Brunn-Minkowski-type inequalities for symmetric convex sets with respect to other probability measures, and obtain several estimates, some of which rely on estimates for poincare constants of restrictions of isotropic log-concave measures to symmetric convex sets.

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