Artem Zvavitch: Volume product and Mahler conjecture for convex bodies

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 $K^{\ast }$. The Mahler conjecture is related to this problem and it asks for the minimum of the volume product $vol\left(K\right)vol\left(K^{\ast }\right)$. 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 recently solved by H. Iriyeh, M. Shibata, in dimension 3. The conjecture is open starting from dimension 4. In this talk I will discuss main properties and ideas related to the volume product and a few different approaches to Mahler conjecture. I will also present ideas of a shorter solution for three dimensional case (this is a part of a joint work with Matthieu Fradelizi, Alfredo Hubard, Mathieu Meyer and Edgardo Roldán-Pensado).