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We explore alternative functional or transport-entropy formulations of theBlaschke-Santal{\'o} inequality and of its conjectured counterpart due toMahler. In particular, we obtain new direct and reverse Blaschke-Santal{\'o}inequalities for s-concave functions. We also obtain new sharp symmetrizedtransport-entropy inequalities for a large class of spherically invariantprobability measures, including the uniform measure on the unit Euclideansphere and generalized Cauchy and Barenblatt distributions. Finally, we showthat the Mahler's conjecture is equivalent to some reinforced log-Sobolev typeinequality on the sphere.

Transport-entropy forms of direct and Converseblaschke-Santal{\'o} inequalities