Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions. (English) Zbl 1486.51010
Summary: 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.
MSC:
| 51F30 | Lipschitz and coarse geometry of metric spaces |
| 28A78 | Hausdorff and packing measures |
| 30L05 | Geometric embeddings of metric spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |
| 54E45 | Compact (locally compact) metric spaces |
Keywords:
purely 1-unrectifiable; Radon-Nikodým property; Whitney arc; Lipschitz-free space; locally flat Lipschitz functionReferences:
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