Two-dimensional B\l{}ocki, $L^p$-Mahler, and Bourgain conjectures

Abstract

We confirm, in dimension two, Blocki's conjectures on sharp lower bounds for Bergman kernels of tube domains. To that end, we verify a broader class of $L^p$-Mahler conjectures due to Berndtsson and the authors, where $p=1$ are Blocki's conjecture, and $p=\infty$ are Mahler's conjectures. The proofs are technically challenging as the $L^p$-Mahler volume is considerably harder to deal with analytically compared to Mahler's volume, and furthermore duality is lost. In addition, unlike in the classical Mahler setting, the non-symmetric setting is considerably more involved than the symmetric one. The proofs involve studying the effect of Mahler's classical sliding of vertices on two-dimensional polytopes on the $L^p$-polar body (no longer a polytope). Some arguments are inspired by works of Campi--Gronchi and Meyer--Reisner on volumes of classical polar bodies of shadow systems. In passing, we also explore how Mahler's sliding affects the isotropic constant. This leads to an elementary proof of Bourgain's strong hyperplane conjectures in dimension two, originally due to Bisztriczky--B\"or\"oczky, Campi--Colesanti--Gronchi and Meckes. Specifically, we show that, as a function of the sliding parameter, the isotropic constant raised to an appropriate power is a convex quadratic polynomial.