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Geometry Seminar: Galyna Livshitz, Georgia Tech

Room 304 Boyd Graduate Studies Bldg.


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.

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