Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

Abstract

[EN] We characterize compact metric spaces whose locally flat Lip-schitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free space F(M) over a compact metric space M is a dual space if and only if M is purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodym property (RNP) and deduce that, for any complete metric space M, pure 1-unrectifiability is actually equivalent to some well-known Banach space properties of F(M) such as the RNP and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the RNP. Finally, we provide a possible solution to a problem of Whitney by finding a rectifiability-based description of 1-critical compact metric spaces, and we use this description to prove the following: a bounded turning tree fails to be 1-critical if and only if each of its subarcs has sigma-finite Hausdorff 1-measure.