Transport Proofs of Some Functional Inverse Santaló Inequalities

In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse Santaló inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with elementary correlation inequalities, we prove these sharp Entropy-Transport inequalities in dimension 1, which therefore gives an alternative transport proof of the sharp functional Mahler conjecture in dimension 1, for both the symmetric and the general case. We also revisit the proof of the functional inverse Santaló inequalities in the n dimensional unconditional case using these transport ideas.