Convex Floating Bodies as Approximations of Bergman Sublevel Sets on Tube Domains

TL;DR

This paper establishes a relationship between Bergman kernel sublevel sets and floating bodies of convex bases in pseudoconvex tube domains, introducing a new affine invariant for convex bodies.

Contribution

It provides the first known estimates linking Bergman kernel sublevel sets to floating bodies, creating a novel affine invariant for convex bodies.

Findings

Establishes estimates relating Bergman sublevel sets to floating bodies

Introduces a new affine invariant for convex bodies

Connects complex analysis with convex geometry

Abstract

For a pseudoconvex tube domain, we prove estimates that relate the sublevel sets of its diagonal Bergman kernel to the floating bodies of its convex base. This allows us to associate a new affine invariant to any convex body.