Azinberg exhaustion problem for bounded complex-convex domains

Determine whether every bounded \(\mathbb{C}\)-convex domain admits an increasing exhaustion by smooth strictly \(\mathbb{C}\)-convex domains.

Background

A bounded C\mathbb{C}-convex domain is a domain whose intersection with every affine complex line is either empty or connected and simply connected. The paper studies smooth and strongly linearly convex exhaustions of such domains and proves an affirmative result under a modulus-of-continuity condition on the outward unit normal, thereby resolving the problem only for a specified regularity class rather than for all bounded C\mathbb{C}-convex domains.

The question originates with Azinberg and concerns approximation from inside by increasingly larger smooth strictly C\mathbb{C}-convex domains. The unrestricted version remains unresolved in the context described by the paper.

References

The genesis of this problem goes back to Azinberg , where he posed the following question: \emph{Does every bounded $\mathbb{C}$-convex domain admit an increasing exhaustion by smooth strictly $\mathbb{C}$-convex domains?}

— Strongly linearly convex exhaustion of a class of $\mathbb{C}$-convex domains  (2609.35474 - Gupta, 28 Sep 2026) in Section 1, Introduction

Another strong form of this quesiton was considered by Pflug and Zwonek, where they asked if every bounded $\mathbb{C}$-convex domain can be exhausted by strongly linearly convex domains and answered it positively for the symmetrized bidisc .

— Strongly linearly convex exhaustion of a class of $\mathbb{C}$-convex domains  (2609.35474 - Gupta, 28 Sep 2026) in Section 1, Introduction