Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T06:48:04.039Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on a Residual Subset of Lipschitz Functions on Metric Spaces

Published online by Cambridge University Press:  30 December 2014

Fabio Cavalletti
Fabio Cavalletti*
Affiliation:
Rheinisch-Westfälische Technische Hochschule Aachen University, Department of Mathematics, Templergraben 64, 52062 Aachen, Germany ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of real-valued Lipschitz functions with non-zero pointwise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous to a result proved for real-valued Lipschitz maps defined on ℝ2 by Alberti et al.

Keywords

Lipschitz functionsnon-zero gradientresidual setsPrimary 53C2330Lxx
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 631 - 636
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alberti, G., Bianchini, S. and Crippa, G., Structure of level sets and Sard-type properties of Lipschitz maps, Annali Scuola Norm. Sup. Pisa 12(4) (2013), 863–902.Google Scholar
2.Burago, D., Burago, Y. and Ivanov, S., A course in metric geometry, Graduate Studies in Mathematics, Volume 33 (American Mathematical Society, Providence, RI, 2001).Google Scholar
3.Cavalletti, F., Decomposition of geodesics in the Wasserstein space and the globalization property, Geom. Funct. Analysis 24(2 (2014), 493–551.CrossRefGoogle Scholar
4.Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Analysis 9 (1999), 428–517.Google Scholar
5.Durand-Cartegna, E. and Jaramillo, J. A., Pointwise Lipschitz functions on metric spaces, J. Math. Analysis Applic. 363 (2010), 525–548.Google Scholar
6.Kleiner, B. and Mackay, J., Differentiable structures on metric measure spaces: a primer, preprint (arXiv:1108.1324, 2011).Google Scholar