Lp-polarity, Mahler volumes, and the isotropic constant

Abstract

This article introduces $L^p$ versions of the support function of a convex body $K$ and associates to these canonical $L^p$-polar bodies $K^{\circ ,p}$ and Mahler volumes ${\mathcal{ℳ}}_p(K)$. Classical polarity is then seen as $L^{\infty }$-polarity. This one-parameter generalization of polarity leads to a generalization of the Mahler conjectures, with a subtle advantage over the original conjecture: conjectural uniqueness of extremizers for each $p\in (0,\infty )$. We settle the upper bound by demonstrating the existence and uniqueness of an $L^p$-Santaló point and an $L^p$-Santaló inequality for symmetric convex bodies. The proof uses Ball’s Brunn–Minkowski inequality for harmonic means, the classical Brunn–Minkowski inequality, symmetrization, and a systematic study of the ${\mathcal{ℳ}}_p$ functionals. Using our results on the $L^p$-Santaló point and a new observation motivated by complex geometry, we show how Bourgain’s slicing conjecture can be reduced to lower bounds on the $L^p$-Mahler volume coupled with a certain conjectural convexity property of the logarithm of the Monge–Ampère measure of the $L^p$-support function. We derive a suboptimal version of this convexity using Kobayashi’s theorem on the Ricci curvature of Bergman metrics to illustrate this approach to slicing. Finally, we explain how Nazarov’s complex-analytic approach to the classical Mahler conjecture is instead precisely an approach to the $L^1$-Mahler conjecture.