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On the dimensional Brunn-Minkowski inequality

Banff International Research Station for Mathematical Innovation and Discovery

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. This is a joint work with A. Kolesnikov.

Mathematics; Functional analysis; Convex and discrete geometry

video/mp4

Author affiliation: Georgia Institute of Technology

2018-09-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67234

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/