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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 forsymmetric convex bodies and made some suggestions on how his proof shouldextend to general convex bodies. This article achieves this extension andserves simultaneously as an exposition to Nazarov's work. A key new ingredientis an affine invariant associated to the Bergman kernel of a tube domain. Thisgives the first `complex' proof of the Bourgain--Milman theorem for generalconvex bodies, specifically, without using symmetrization.

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