Geometric and Functional Analysis

, Volume 28, Issue 6, pp 1641–1705 | Cite as

Sharp Finiteness Principles For Lipschitz Selections

  • Charles Fefferman
  • Pavel ShvartsmanEmail author


Let \({(\mathcal{M}, \rho) }\) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from \({\mathcal{M}}\) into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness constant.

Keywords and phrases

Set-valued mapping Lipschitz selection Metric tree Helly’s theorem Nagata dimension Whitney partition Steiner-type point 

Mathematics Subject Classification



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We are grateful to Alexander Brudnyi, Arie Israel, Bo’az Klartag, Garving (Kevin) Luli and the participants of the 10th Whitney Problems Conference, Williamsburg, VA, for valuable conversations. We thank the referee for very careful reading and numerous suggestions, which led to improvements in our exposition. We are grateful also to the College of William and Mary, Williamsburg, VA, USA, the American Institute of Mathematics, San Jose, CA, USA, the Fields Institute, Toronto, Canada, the University of Arkansas, AR, USA, the Banff International Research Station, Banff, Canada, the Centre International de Rencontres Mathématiques (CIRM), Luminy, Marseille, France, and the Technion, Haifa, Israel, for hosting and supporting workshops on the topic of this paper and closely related problems. Finally, we thank the US-Israel Binational Science Foundation, the US National Science Foundation, the Office of Naval Research and the Air Force Office of Scientific Research for generous support.


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Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

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