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Tuesday, September 3, 2013

**Abstract:** Let K be a centrally symmetric convex body in R^n, and let K* be its dual, that is, K* consists of all y in R^n such that for all x in K the inner product xy<1. The quantity M(K)=vol(K)vol(K*) is called Maler's volume product. Mahler (1939) conjectured that the minimum of M(K) is attained on the cube Q. Mahler proved it only for n=2; for n>2 it is still open. Bourgain and Milman (1987) proved the estimate M(K) > c^n M(Q), in which the best constant was obtained by Kuperberg (2008). I present the amazing proof by Nazarov (2012) based on Bergman kernel and Hormander's estimates of d-bar equations.