L P-POLARITY, MAHLER VOLUMES, AND THE ISOTROPIC CONSTANT

Journal article, 2024

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 degrees ;p and Mahler volumes M p (K). Classical polarity is then seen as L 1-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 2 (0, 1 ). We settle the upper bound by demonstrating the existence and uniqueness of an L p-Santal & oacute; point and an L p-Santal & oacute; 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 M p functionals. Using our results on the L p-Santal & oacute; 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 & egrave;re 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.