Near geodesics in John domains in Banach spaces

TL;DR

This paper proves that in certain Banach space domains, near geodesics are cone arcs when the domain is a John domain homeomorphic to an inner uniform domain via a CQH map.

Contribution

It establishes a new geometric property of near geodesics in Banach space domains under specific homeomorphism conditions.

Findings

Near geodesics in John domains are cone arcs.

The result applies to domains homeomorphic to inner uniform domains via CQH maps.

Advances understanding of geometric structures in Banach spaces.

Abstract

Let $E$ be a real Banach space with dimension at least 2. In this paper, we prove that if $D\subset E$ is a John domain which is homeomorphic to an inner uniform domain via a CQH map, then each neargeodesic in $D$ is a cone arc.