Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

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-Nikodým 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.

v2: Added Section 1.3 introducing Kirchheim's theorems on rectifiability

References in corpus (3)

• Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon Nikodym property
• Free spaces over some proper metric spaces
• Bi-Lipschitz geometry of quasiconformal trees

Cited by in corpus (7)

• Residuality in the set of norm attaining operators between Banach spaces
• Extreme points in Lipschitz-free spaces over compact metric spaces
• Delta-points and their implications for the geometry of Banach spaces
• De Leeuw representations of functionals on Lipschitz spaces
• Hyperbolic Metric Spaces and Stochastic Embeddings
• Projections in Lipschitz-free spaces induced by group actions
• Tilings of the Hyperbolic Space and Lipschitz Functions