Equivalent characterizations of John and uniform domains in doubling metric spaces

Abstract

In this paper, we characterize John and uniform domains in doubling metric spaces. Specifically, we show that a locally quasiconvex domain in a doubling metric space is length John if and only if it is diameter John. For uniform domains, we prove that a domain in a doubling metric space is length uniform if and only if it is diameter uniform (or distance uniform) and locally quasiconvex. Moreover, in a doubling length metric space, we refine this result by showing that a domain is length uniform (resp. John) if and only if it is diameter uniform (resp. John).