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

Revista Matemática Iberoamericana (1996)

- Volume: 12, Issue: 2, page 337-410
- ISSN: 0213-2230

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topSemmes, Stephen. "On the nonexistence of bilipschitz parametrizations and geometric problems about A∞-weights.." Revista Matemática Iberoamericana 12.2 (1996): 337-410. <http://eudml.org/doc/39505>.

@article{Semmes1996,

abstract = {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 Rn 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 wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bilipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on Rn using an A∞ weight.},

author = {Semmes, Stephen},

journal = {Revista Matemática Iberoamericana},

keywords = {Aplicación Lipschitziana; Espacios métricos; Variedades topológicas; Espacio euclídeo; Equivalencias geométricas; Parametrización; bi-Lipschitz equivalent to a Euclidean space; bi-Lipschitz parameterizations; deforming the Euclidean metric; weight},

language = {eng},

number = {2},

pages = {337-410},

title = {On the nonexistence of bilipschitz parametrizations and geometric problems about A∞-weights.},

url = {http://eudml.org/doc/39505},

volume = {12},

year = {1996},

}

TY - JOUR

AU - Semmes, Stephen

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

JO - Revista Matemática Iberoamericana

PY - 1996

VL - 12

IS - 2

SP - 337

EP - 410

AB - 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 Rn 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 wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bilipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on Rn using an A∞ weight.

LA - eng

KW - Aplicación Lipschitziana; Espacios métricos; Variedades topológicas; Espacio euclídeo; Equivalencias geométricas; Parametrización; bi-Lipschitz equivalent to a Euclidean space; bi-Lipschitz parameterizations; deforming the Euclidean metric; weight

UR - http://eudml.org/doc/39505

ER -

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- Pierre Pansu, Plongements quasiisométriques du groupe de Heisenberg dans ${L}^{p}$, d’après Cheeger, Kleiner, Lee, Naor
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