The Nazarov proof of the non-symmetric Bourgain--Milman inequality

TL;DR

This paper extends Nazarov's complex-analytic proof of the Bourgain--Milman inequality from symmetric to general convex bodies, introducing a new affine invariant related to Bergman kernels.

Contribution

It provides the first complex proof of the Bourgain--Milman inequality for all convex bodies, avoiding symmetrization, and introduces a novel affine invariant based on Bergman kernels.

Findings

First complex proof for general convex bodies

Introduction of a new affine invariant

Extension of Nazarov's method to non-symmetric bodies

Abstract

In 2012, Nazarov used Bergman kernels and Hormander's $L^2$ estimates for the $\bar\partial$-equation to give a new proof of the Bourgain--Milman theorem for symmetric convex bodies and made some suggestions on how his proof should extend to general convex bodies. This article achieves this extension and serves simultaneously as an exposition to Nazarov's work. A key new ingredient is an affine invariant associated to the Bergman kernel of a tube domain. This gives the first `complex' proof of the Bourgain--Milman theorem for general convex bodies, specifically, without using symmetrization.