**Date and time:**

**Speaker: **Galyna Livshitz, Georgia Tech

**Title: ***On the role of symmetry in isoperimetric-type inequalities *

**Abstract: **In the recent y ears, 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 Rn are in fact 1 - concave with respect to the addition of symmetric convex n sets. In this talk we shall prove that the standard Gaussian measure enjoys 1 concavity with respect to 2n centered convex sets. The 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 indeed better-than-log-concave with respect to the addition of symmetric convex sets. This is a joint work with A. Kolesnikov.