Strongly linearly convex exhaustion of a class of $\mathbb{C}$-convex domains

Abstract

Let $D\subseteq {\mathbb{C}}^n$ be a bounded $\mathbb{C}$-convex domain with $C^1$ boundary whose outward unit normal admits a modulus of continuity $\omega$ satisfying ${\lim }_{t\to 0^{+}}\frac{\omega (t)}{\sqrt{t}}$. We prove that $D$ admits an increasing exhaustion by bounded $C^{\infty }$ strongly linearly convex domains. This, in particular, answers a question posed by Azinberg \cite{azin} in affirmative for a class of domains $D$.

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