Transport-entropy forms of direct and Converseblaschke-Santal{ó} inequalities

We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{ó} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{ó} inequalities for s-concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions. Finally, we show that the Mahler’s conjecture is equivalent to some reinforced log-Sobolev type inequality on the sphere.