From Bergman kernels to Polarity: Perspectives on the Mahler, Bourgain, and Blocki conjectures

Abstract

This thesis investigates fundamental questions about convex bodies, polarity, and volume. By revisiting their connection to Bergman kernels, novel concepts of $L^p$-polarity and Mahler volumes are introduced and studied extensively. This elucidates Nazarov's approach to the Mahler conjecture and simultaneously offers a new approach both to the Mahler Conjectures and Bourgain's Hyperplane Conjecture. Moreover, it unifies B\l{}ocki's and Mahler's Conjectures as the endpoint cases of new $L^p$-Mahler Conjectures. Key contributions include refined bounds, new inequalities, the resolution of B\l{}ocki's conjecture for Bergman kernels of tube domains in dimension two, and the development of new links between the Mahler conjecture and Bourgain's conjecture. In the first part of this thesis (Chapter 2), Nazarov's bound on the Bergman kernels of tube domains is revisited. A measure of asymetry for convex bodies is introduced, and Nazarov's proof of the Bergman kernel bound is reformualted to avoid the use of John's theorem and remove the symmetry assumption for convex bodies. A new bound for Bergman kernels of tube domains is developed, tailored to the symmetry of the body. This approach recovers Nazarov's bound for symmetric bodies and establishes a new bound applicable to all convex bodies. Motivated by the study of Bergman kernels of tube domains, the third chapter introduces the concept of $L^p$-polarity, a novel notion that provides a smooth approximation of classical polarity. $L^p$-polarity is studied in detail; the existence and uniqueness of $L^p$-Santal'o points along with $L^p$-Santal'o inequalities, both in the symmetric and non-symmetric cases, are established. Steiner symmetrization is used as the main tool in proving the $L^p$-Santal'o inequalities. Using these notions, Bergman kernels of tube domains are now understood via the volume of the $L^1$-polar body, leading to the derivation of sharp upper bounds as a corollary. Additionally, the $L^p$-Mahler volumes and Santal'o points are utilized in the derivation of an upper bound on the isotropic constant, leading to a novel approach to Bourgain's slicing problem. The introduction of $L^p$-Mahler volumes naturally leads to the formulation of $L^p$-Mahler conjectures. Numerical evidence is provided to suggest that, unlike classical polarity, the cube and its linear images are the unique minimizers in the symmetric case. In Chapter 4, the $L^p$-Mahler conjectures are verified in dimension two by adapting Mahler's sliding argument both in the symmetric and non-symmetric cases. In the latter case, triangles centered at the origin are proven to be the minimizers. As a corollary, sharp lower bounds on the Bergman kernels of tube domains are established in dimension two, verifying in the affirmative a conjecture of B\l{}ocki in that case. Lastly, $L^p$-Legendre transforms are introduced as the functional analogues of $L^p$-polarity. Functional $L^p$-Santal'o points and $L^p$-Santal'o inequalities are established. This is achieved using the Fokker--Planck heat flow method introduced by Nakamura--Tsuji. Additionally, alternative approaches inspired by the work of Artstein--Klartag--Milman are explored by computing the asymptotics in dimension of the $L^p$-Mahler volumes of the Euclidean ball.