Bi-Lipschitz approximation by finite-dimensional imbeddings

Karin Usadi Katz; Mikhail G. Katz

doi:10.1007/s10711-010-9497-4

Bi-Lipschitz approximation by finite-dimensional imbeddings

Karin Usadi Katz, Mikhail G. Katz

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

Gromov's celebrated systolic inequality from '83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1+ε)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every ε > 0, using the first variation formula and the mean value theorem.

Original language	English
Pages (from-to)	131-136
Number of pages	6
Journal	Geometriae Dedicata
Volume	150
Issue number	1
DOIs	https://doi.org/10.1007/s10711-010-9497-4
State	Published - Feb 2011

Keywords

Essential manifold
Finite-dimensional approximation
First variation formula
Geodesic
Gromov's inequality
Infinitesimal
Injectivity radius
Kuratowski imbedding
Systole
Systolic inequality

All Science Journal Classification (ASJC) codes

Geometry and Topology

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10.1007/s10711-010-9497-4

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title = "Bi-Lipschitz approximation by finite-dimensional imbeddings",

abstract = "Gromov's celebrated systolic inequality from '83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1+ε)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every ε > 0, using the first variation formula and the mean value theorem.",

keywords = "Essential manifold, Finite-dimensional approximation, First variation formula, Geodesic, Gromov's inequality, Infinitesimal, Injectivity radius, Kuratowski imbedding, Systole, Systolic inequality",

author = "Katz, \{Karin Usadi\} and Katz, \{Mikhail G.\}",

note = "Funding Information: Supported by the Israel Science Foundation (grants no. 84/03 and 1294/06) and the BSF (grant 2006393).",

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doi = "10.1007/s10711-010-9497-4",

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AU - Katz, Karin Usadi

AU - Katz, Mikhail G.

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N2 - Gromov's celebrated systolic inequality from '83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1+ε)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every ε > 0, using the first variation formula and the mean value theorem.

AB - Gromov's celebrated systolic inequality from '83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1+ε)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every ε > 0, using the first variation formula and the mean value theorem.

KW - Essential manifold

KW - Finite-dimensional approximation

KW - First variation formula

KW - Geodesic

KW - Gromov's inequality

KW - Infinitesimal

KW - Injectivity radius

KW - Kuratowski imbedding

KW - Systole

KW - Systolic inequality

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DO - 10.1007/s10711-010-9497-4

M3 - مقالة

SN - 0046-5755

VL - 150

SP - 131

EP - 136

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 1

ER -

Bi-Lipschitz approximation by finite-dimensional imbeddings

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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