  We explore alternative functional or transport-entropy formulations of the
Blaschke-Santal{\'o} inequality and of its conjectured counterpart due to
Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{\'o}
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.
