2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116578We 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 K. Ball 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, Klartag and Milman, with its equality case.19 pages, to appear in Mathematische ZeitschriftFunctional AnalysisMetric Geometry52A40text