Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

TL;DR

This paper characterizes purely 1-unrectifiable metric spaces through locally flat Lipschitz functions, linking geometric properties to Banach space features and solving longstanding problems in Lipschitz analysis and rectifiability.

Contribution

It provides a geometric characterization of purely 1-unrectifiable spaces and connects this to Banach space properties, addressing open questions in Lipschitz analysis.

Findings

Purely 1-unrectifiable spaces are characterized by locally flat Lipschitz functions.

The Lipschitz-free space over such spaces is a dual space.

Pure 1-unrectifiability is equivalent to RNP and Schur property in the free space.

Abstract

We characterize compact metric spaces whose locally flat Lipschitz 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 $\mathcal{F}(M)$ 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-Nikod\'ym 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 $\mathcal{F}(M)$ 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.