For any compact metric space M , we prove that the locally flat Lipschitz functions separate points (of M ) uniformly if and only if M is purely 1-unrectifiable, resolving a problem posed by Weaver in 1999. 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 1unrectifiable. Furthermore, for any complete metric space M , we deduce that pure 1-unrectifiability actually characterizes some well-known Banach space properties of F (M ) such as the Radon-Nikodým property and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the Radon-Nikodým property. Finally, we provide a solution to a problem of Whitney from 1935 by finding a rectifiability-based characterization of 1-critical compact metric spaces, and we use this characterization to prove the following: a bounded turning tree fails to be 1-critical if and only each of its subarcs has σ-finite Hausdorff 1-measure.
Abstract. We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.
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