An origin symmetric convex body K in R^n can always be realized as the unit ball of a norm. The unit ball of the dual norm K* is called the dual body, or polar body, of K. The volume product P(K) is an affine invariant of K defined by P(K)=vol(K)vol(K*). It was shown by Blaschke (for low dimensions) and Santalo (in general) that P(K) is maximized when K is an ellipsoid. It was proven much later that ellipsoids are the only bodies that maximize the volume product. In this talk we will describe a proof of these facts using Fourier analysis. This is joint work with Gabriele Bianchi.