Description
Abstract: The Brunn-Minkowski inequality states that for (open) sets A and B in R^d, we have |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}. Equality holds if and only if A and B are convex and homothetic sets in R^d. In this talk, we present a sharp stability result for the Brunn-Minkowski inequality, concluding a long line of research on this problem. We show that if we are close to equality in the Brunn-Minkowski inequality, then A and B are close to being homothetic and convex, establishing the exact dependency between the three notions of closeness. This is based on joint work with Alessio Figalli, Peter van Hintum, and Marius Tiba.