Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property

By Jeff Cheeger, Bruce Kleiner. 2008

86 citations recorded by the source corpus.

We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon–Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/b4ea5869d477d2ee041bd4b00474c86bc1b6e035

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

On the differential structure of metric measure spaces and applications (2012, methodology): “3 in [17] obtained by Cheeger and Kleiner, where the span of the set of derivations along curves is shown to be dense in the tangent space under doubling and Poincaré assumptions.”
Metric differentiation, monotonicity and maps to L1 (2009, methodology): “In this paper, we continue our investigation of Lipschitz maps of metric spaces into Banach spaces [9, 11, 12 ], which is motivated by the role of bi-Lipschitz embedding problems in theoretical computer science [4, 26, 28, 29], earlier developments in the infinitesimal geometry of metric measure sp”