Rigid characterizations of pseudoconvex domains

TL;DR

This paper characterizes pseudoconvex domains in complex space by examining the pseudoconvexity of largest balanced subdomains centered at each point, and explores similar characterizations for linearly convex domains.

Contribution

It provides a new characterization of pseudoconvex domains based on balanced subdomains, extending to linearly convex domains.

Findings

Pseudoconvexity of a domain is equivalent to the pseudoconvexity of all largest balanced subdomains.

Analogues of the characterization are established for linearly convex domains.

Abstract

We prove that an open set $D$ in $\C^n$ is pseudoconvex if and only if for any $z\in D$ the largest balanced domain centered at $z$ and contained in $D$ is pseudoconvex, and consider analogues of that characterization in the linearly convex case.