Analysis Seminar: "On Mahler conjecture for convex bodies"

Abstract: Let K be convex, symmetric, with respect to the origin, body in R^n. One of the major open problems in convex geometry is to understand the connection between the volumes of K and the polar body K^∗. The Mahler conjecture is related to this problem and it asks for the minimum of the volume product vol(K)vol(K^* ). In 1939, Santalo proved that the maximum of the volume product is attained on the Euclidean ball. About the same time Mahler conjectured that the minimum should be attained on the unit cube or its dual - cross-polytope. Mahler himself proved the conjectured inequality in R^2.  The question was very recently solved by H. Iriyeh, M. Shibata, in dimension 3. The conjecture is open staring from dimension 4.  In this talk I will discus  a few different approaches to Mahler conjecture and the volume product in general. L will also present a recent short solution for three dimensional case.

Host: Brett Wick