I am aware of a theorem that goes as follows: If $X$ is a compact, connected metric space, then the metric on $X$ is equivalent to a convex metric if and only if $X$ is locally connected. Here, locally connected means every point has a local basis of open, connected neighborhoods - a convex metric means a metric so that for every pair of distinct points $x, y \in X$, if $d(x,y)=2r$ there is a $z \in X$ with $d(x,z) = d(y,z) = r$.
One direction is elementary, and is given here, namely that a convex continuum is locally connected: Convex metric on a contiuum.
The proof of the other direction was first exhibited by Moise: https://projecteuclid.org/euclid.bams/1183514376
Does anyone know if this theorem has appeared in book form before, preferably in a more modern setting? Thanks a lot!
