A Santaló inequality for the L^p-polar body

2024/01/19 by Mastrantonis, Vlassis
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2401.10836

Abstract

In recent work with Berndtsson and Rubinstein, a notion of L^p-polarity was introduced, with classical polarity recovered in the limit p→∞, and L¹-polarity closely related to Bergman kernels of tube domains. A Santaló 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-Santaló inequality holds for any convex body after translation by the L^p-Santaló 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.

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