Some functional forms of Blaschke–Santaló inequality

We establish new functional versions of the Blaschke–Santaló inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of Ball Isometric problems in ℓ _p and sections of convex sets. PhD Dissertation, Cambridge, 1986] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for log-concave functions and Legendre transforms which extend the recent result of Artstein et al. Mathematika 51:33–48, 2004], with its equality case.