No abstract

Mathematics Subject Classification. Primary 30E99; Secondary 30H50.

In 2012, Nazarov used Bergman kernels and Hörmander's L 2 estimates for the ∂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.