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Mathematische Annalen

, Volume 359, Issue 3–4, pp 595–628 | Cite as

A general theorem of existence of quasi absolutely minimal Lipschitz extensions

  • Matthew J. Hirn
  • Erwan Y. Le Gruyer
Article
  • 158 Downloads

Abstract

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier “quasi” to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.

Mathematics Subject Classification (1991)

54C20 58C25 46T20 49-XX 39B05 

Notes

Acknowledgments

E.L.G. is partially supported by the ANR (Agence Nationale de la Recherche) through HJnet projet ANR-12-BS01-0008-01. M.J.H. would like to thank IRMAR (The Institue of Research of Mathematics of Rennes) for supporting his visit in 2011, during which time the authors laid the foundation for this paper. Both authors would like to acknowledge the Fields Institute for hosting them for two weeks in 2012, which allowed them to complete this work. Both authors would also like to thank the anonymous reviewer for his or her helpful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Département d’InformatiqueÉcole normale supérieureParisFrance
  2. 2.INSA de Rennes & IRMARRennes Cedex 7France

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