"Convex" characterization of linearly convex domains

TL;DR

This paper characterizes linearly convex domains in complex space using a geometric condition involving the convex hulls of pairs of discs with common centers.

Contribution

It provides a new geometric characterization of linearly convex domains in terms of convex hulls of discs, linking smoothness and convexity properties.

Findings

A $C^{1,1}$-smooth bounded domain is linearly convex iff convex hulls of centered discs are contained within the domain.

Establishes an equivalence between linear convexity and a geometric condition involving discs.

Enhances understanding of convexity properties in complex analysis.

Abstract

We prove that a $C^{1,1}$-smooth bounded domain $D$ in $\C^n$ is linearly convex if and only if the convex hull of any two discs in $D$ with common center lies in $D.$