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 discuss a few different approaches to Mahler conjecture and the volume product in general. I will also present a short solution for three dimensional case which was created in the joint work with my friends from LAMA (this is a joint work with Matthieu Fradelizi, Alfredo Hubard, Mathieu Meyer and Edgardo Roldán-Pensado).
