Uniform Lipschitz-connectedness and metric convexity - Paranjothi Pillay; Dharmanand Baboolal (Categories and General Algebraic Structures with Applications)
In this paper we continue with our study of uniformly Lipschitz-connected metric spaces.   We obtain further properties of uniformly Lipschitz-connected metric spaces and then obtain a generalisation of a result due to Edelstein.  In addition, we show that for a proper Lipschitz-connected metric space,  $L_d = 1$ precisely when $X$ is convex, which leads us to conjecture that $L_d$ is a kind of measure of convexity in a proper Lipschitz-connected metric space.  We provide some examples to corroborate our conjecture.