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Geometriae Dedicata

, Volume 150, Issue 1, pp 131–136 | Cite as

Bi-Lipschitz approximation by finite-dimensional imbeddings

  • Karin Usadi Katz
  • Mikhail G. KatzEmail author
Original Paper

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 \({(1+\epsilon)}\)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every \({\epsilon > 0}\), 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 

Mathematics Subject Classification (2000)

Primary 53C23 Secondary 26E35 

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael

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