Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality

TL;DR

This paper extends the characterization of John domains to higher dimensions using metric duality and introduces new properties like LLC and homological bounded turning, revealing limitations in 3D cases.

Contribution

It generalizes the characterization of John domains in higher dimensions and demonstrates a counterexample in three dimensions affecting homotopic bounded turning.

Findings

Extended characterization of John domains in higher dimensions.

Constructed a 3D uniform domain with unique topological properties.

Showed limitations of homotopic bounded turning characterization in 3D.

Abstract

In this paper, we extend the characterization of John disks obtained by N\"akki and V\"ais\"al\"a [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by V\"ais\"al\"a in his study of metric duality theory [Math. Scan. 1997]. Somewhat surprisingly, we constructed a uniform domain in $\R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three.