$L^p$-polarity, Mahler volumes, and the isotropic constant

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 $L^p$-Santal\'o point and an $L^p$-Santal\'o 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 $L^p$-Santal\'o 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--Amp\`ere 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.