Equivalent characterizations of John and uniform domains in doubling metric spaces

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).