Inner uniform domains and the Apollonian inner metric

TL;DR

This paper characterizes inner uniform domains in Euclidean space using Apollonian inner metrics and introduces a new characterization for A-uniform domains, advancing the understanding of geometric domain properties.

Contribution

It provides a novel characterization of inner uniform and A-uniform domains via Apollonian metrics, linking geometric domain properties with specific metric conditions.

Findings

Characterization of inner uniform domains using Apollonian inner metric.

New criteria for A-uniform domains based on metric properties.

Enhanced understanding of geometric domain classifications.

Abstract

In this paper, we characterize inner uniform domains in $\IR^n$ in terms of Apollonian inner metric and the metric $j'_D$ when $D$ are Apollonian. As an application, a new characterization for $A$-uniform domains is obtained.