Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

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Gâteaux differentiability of Lipschitz functions

Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) ◽

10.23943/princeton/9780691153551.003.0002 ◽

2012 ◽

Author(s):

Joram Lindenstrauss ◽

David Preiss ◽

Tiˇser Jaroslav

Keyword(s):

Existence Result ◽

Mean Value ◽

Lipschitz Functions ◽

Gâteaux Differentiability ◽

Lipschitz Maps ◽

Locally Lipschitz ◽

Nikodym Property ◽

The Mean ◽

Mere Existence ◽

Gateaux Differentiability

This chapter presents the main results on Gâteaux differentiability of Lipschitz functions by recalling the notions of the Radon-Nikodým property (RNP) and null sets. The discussion focuses not only on the mere existence of points of Fréchet differentiability, but also, and often more important, on the validity of the mean value estimates. After considering the RNP of a Banach space, the chapter examines Haar and Aronszajn-Gauss null sets. It then analyzes the existence result for Gâteaux derivatives as well as the meaning of multidimensional mean value estimates. It also explains how, for locally Lipschitz maps of separable Banach spaces to spaces with the RNP, the condition for the validity of the multidimensional mean value estimate may be simplified.

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091514000261 ◽

2014 ◽

Vol 58 (3) ◽

pp. 631-636

Author(s):

Fabio Cavalletti

Keyword(s):

Banach Space ◽

Probability Measure ◽

Metric Spaces ◽

Lipschitz Constant ◽

Lipschitz Functions ◽

Residual Subset ◽

Lipschitz Maps ◽

Almost Everywhere ◽

Bounded Functions ◽

Separable Metric Space

AbstractLet (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.

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