### On the dimensional Brunn-Minkowski inequality

Probability Seminar

4th May 2018, 3:30 pm – 4:30 pm

Main Maths Building, SM4

In the recent years, a number of conjectures has appeared, concerning the improvement of the inequalities of Brunn-Minkowski type under the additional assumptions of symmetry; this includes the B-conjecture, the Gardner-Zvavitch conjecture of 2008, the Log-Brunn-Minkowski conjecture of 2012, and some variants. The conjecture of Gardner and Zvavitch, also known as dimensional Brunn-Minkowski conjecture, states that even log-concave measures in $\R^n$ are in fact $\frac{1}{n}$- concave with respect to the addition of symmetric convex sets. In this talk we shall establish the validity of the Gardner-Zvavitch conjecture asymptotically, and prove that the standard Gaussian measure enjoys $\frac{0.37}{n}$ concavity with respect to centered convex sets. Some improvements to the case of general log-concave measures shall be discussed as well: under certain compactness assumption, we show that any symmetric log-concave measure is better-than-log-concave with respect to the addition of symmetric convex sets. This is a joint work with A. Kolesnikov.

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