Displaying similar documents to “Some remarks about metric spaces, spherical mappings, functions and their derivatives.”

Bilipschitz embeddings of metric spaces into euclidean spaces.

Stephen Semmes (1999)

Publicacions Matemàtiques

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When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small ("snowflake") deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, like rectifiability, differentiability, and curves of finite length. Here we discuss a (somewhat...

On the nonexistence of bilipschitz parametrizations and geometric problems about A-weights.

Stephen Semmes (1996)

Revista Matemática Iberoamericana

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How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of R are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parametrizations are...

Gehring's lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals

Bruno Franchi, Francesco Serra Cassano (1996)

Studia Mathematica

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We prove a higher integrability result - similar to Gehring's lemma - for the metric space associated with a family of Lipschitz continuous vector fields by means of sub-unit curves. Applications are given to show the higher integrability of the gradient of minimizers of some noncoercive variational functionals.

Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

Jeff Cheeger, Bruce Kleiner, Andrea Schioppa (2016)

Analysis and Geometry in Metric Spaces

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We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14, 27]. As corollarieswe give a new proof of one of the main results of [14], a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith [27] is equal to 1, and new nonembeddability results.

Definitions of Sobolev classes on metric spaces

Bruno Franchi, Piotr Hajłasz, Pekka Koskela (1999)

Annales de l'institut Fourier

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There have been recent attempts to develop the theory of Sobolev spaces W 1 , p on metric spaces that do not admit any differentiable structure. We prove that certain definitions are equivalent. We also define the spaces in the limiting case p = 1 .