We establish new functional versions of the Blaschke-Santaló inequality on the volume product of a convex body which generalize to the nonsymmetric setting an inequality of K. Ball [2] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for logconcave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman [1], with its equality case.
It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non‐symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.
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