A Santal\'o inequality for the $L^p$-polar body

Abstract

In recent work with Berndtsson and Rubinstein, a notion of $L^p$-polarity was introduced, with classical polarity recovered in the limit $p\to\infty$, and $L^1$-polarity closely related to Bergman kernels of tube domains. A Santal\'o inequality for the $L^p$-polar was proved for symmetric convex bodies. The aim of this article is to remove the symmetry assumption. Thus, an $L^p$-Santal\'o inequality holds for any convex body after translation by the $L^p$-Santal\'o point. As a corollary, this yields an optimal upper bound on Bergman kernels of tube domains. The proof is by Steiner symmetrization, but unlike the symmetric case, a careful translation of the body is required before each symmetrization.