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Sharp Finiteness Principles For Lipschitz Selections

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Abstract

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.

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References

  1. Assouad P.: Sur la distance de Nagata. C. R. Acad. Sci. Paris Sér. I Math. 294(1), 31–34 (1982)

    MathSciNet  MATH  Google Scholar 

  2. Brodskiy N., Dydak J., Higes J., Mitra A.: Assouad-Nagata dimension via Lipschitz extensions. Israel J. Math.171, 405–423 (2009)

    Article  MathSciNet  Google Scholar 

  3. Brudnyi A.: A note on the Lipschitz selection. C. R. Math. Rep. Acad. Sci. Canada, 40(1), 29–32 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Brudnyi A., Brudnyi Yu.: Methods of Geometric Analysis in extension and trace problems Vols. I, II, Monographs in Mathematics, Vol. 102. Springer, Basel (2012)

    MATH  Google Scholar 

  5. Y. Brudnyi, and P. Shvartsman. Generalizations of Whitney’s extension theorem. Internat. Math. Res. Notices (3) (1994), 129–139.

  6. Brudnyi Yu., Shvartsman P.: The trace of jet space \({{J}^{k} \Lambda{ }_\omega}\) to an arbitrary closed subset of \({\mathbb{R}^{n}}\) . Trans. Amer. Math. Soc., 350(4), 1519–1553 (1998)

    Article  MathSciNet  Google Scholar 

  7. Brudnyi Yu., Shvartsman P.: Whitney Extension Problem for Multivariate \({{C}^1,^\omega}\) -functions. Trans. Amer. Math. Soc. 353(6), 2487–2512 (2001)

    Article  MathSciNet  Google Scholar 

  8. Callahan P.B., Kosaraju S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. Assoc. for Computing Machinery, 42, 67–90 (1995)

    Article  MathSciNet  Google Scholar 

  9. L. Danzer, B. Grünbaum, and V. Klee. Helly’s Theorem and its relatives. In: AMS Symposium on Convexity, Seattle, Proceedings of Symposium on Pure Mathematics, Vol. 7. Amer. Math. Soc., Providence, RI (1963), pp. 101–180.

  10. Fefferman C.: A sharp form of Whitney extension theorem. Annals of Math., 161(1), 509–577 (2005)

    Article  MathSciNet  Google Scholar 

  11. Fefferman C.: A Generalized Sharp Whitney Theorem for Jets. Rev. Mat. Iberoamericana, 21(2), 577–688 (2005)

    Article  MathSciNet  Google Scholar 

  12. Fefferman C.: Whitney extension problem for C m. Annals of Math. 164(1), 313–359 (2006)

    Article  MathSciNet  Google Scholar 

  13. Fefferman C., Klartag B.: Fitting a C m-smooth function to data I. Annals of Math. 169(1), 315–346 (2009)

    Article  MathSciNet  Google Scholar 

  14. Fefferman C., Klartag B.: Fitting a C m-smooth function to data II. Revista Mat. Iberoamericana, 25(1), 49–273 (2009)

    Article  MathSciNet  Google Scholar 

  15. Fefferman C.: Fitting a C m-smooth function to data III. Annals of Math. 170(1), 427–441 (2009)

    Article  MathSciNet  Google Scholar 

  16. Fefferman C.: Whitney extension problems and interpolation of data. Bulletin A.M.S. 46(2), 207–220 (2009)

    Article  MathSciNet  Google Scholar 

  17. Fefferman C., Israel A., Luli G.K.: Finiteness principles for smooth selection. Geom. Funct. Anal. 26(2), 422–477 (2016)

    Article  MathSciNet  Google Scholar 

  18. C. Fefferman, A. Israel, and G. K. Luli. Finiteness principles for \({{C}^{m}}\) and Lipschitz selection. 82 pp., Whitney Extension Problems: \({{C}^{m}}\) and Sobolev functions on subsets of \({\mathbb{R}^{n}}\) . The 9th Whitney Problems Workshop, May June 2016, Haifa, Israel. http://www.math.technion.ac.il/Site/events/EvntOffices/event.php?eid=eo108. Accessed June 2016.

  19. Fefferman C., Israel A., Luli G.K.: Interpolation of data by smooth non-negative functions. Rev. Mat. Iberoam. 33(1), 305–324 (2017)

    Article  MathSciNet  Google Scholar 

  20. C. Fefferman, and P. Shvartsman. Sharp finiteness principles for Lipschitz selections: long version. arXiv:1708.00811v2.

  21. Har-Peled S., Mendel M.: Fast Construction of Nets in Low– Dimensional Metrics and Their Applications. SIAM J. Comput. 35(5), 1148–1184 (2006)

    Article  MathSciNet  Google Scholar 

  22. U. Lang, and T. Schlichenmaier. Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not., (58) (2005), 3625–3655.

  23. Nagata J.: Note on dimension theory for metric spaces. Fund. Math. 45, 143–181 (1958)

    Article  MathSciNet  Google Scholar 

  24. Ostrovskii M., Rosenthal D.: Metric dimensions of minor excluded graphs and minor exclusion in groups. Internat. J. Algebra Comput. 25(4), 541–554 (2015)

    Article  MathSciNet  Google Scholar 

  25. P. Shvartsman. The traces of functions of two variables satisfying the Zygmund condition. (Russian) Studies in the Theory of Functions of Several Real Variables, 145–168, Yaroslav. Gos. Univ., Yaroslavl’ (1982).

  26. Shvartsman P.: Lipschitz sections of set–valued mappings and traces of functions from the Zygmund class on an arbitrary compactum (Russian). Dokl. Akad. Nauk SSSR, 276(3), 559–562 (1984) English transl. in Soviet Math. Dokl. 29 (1984), no. 3, 565–568

    MathSciNet  Google Scholar 

  27. P. Shvartsman. Lipschitz sections of multivalued mappings. (Russian) Studies in the theory of functions of several real variables, 121–132, 149, Yaroslav. Gos. Univ., Yaroslavl’ (1986).

  28. Shvartsman P.: On the traces of functions of the Zygmund class. Sib. Mat. Zh. 28(5), 203–215 (1987) English transl. in Sib. Math. J. 28 (1987), 853–863.

    MathSciNet  MATH  Google Scholar 

  29. P. Shvartsman. K-functionals of weighted Lipschitz spaces and Lipschitz selections of multivalued mappings. Interpolation spaces and related topics, (Haifa, 1990), 245–268, Israel Math. Conf. Proc., 5, Bar-Ilan Univ., Ramat Gan (1992).

  30. Shvartsman P.: On Lipschitz selections of affine-set valued mappings. Geom. Funct. Anal. 11(4), 840–868 (2001)

    Article  MathSciNet  Google Scholar 

  31. Shvartsman P.: Lipschitz selections of set–valued mappings and Helly’s theorem. J. Geom. Anal. 12(2), 289–324 (2002)

    Article  MathSciNet  Google Scholar 

  32. Shvartsman P.: Barycentric selectors and a Steiner–type point of a convex body in a Banach space. J. Funct. Anal. 210(1), 1–42 (2004)

    Article  MathSciNet  Google Scholar 

  33. Shvartsman P.: The Whitney extension problem and Lipschitz selections of set–valued mappings in jet–spaces. Trans. Amer. Math. Soc. 360(10), 5529–5550 (2008)

    Article  MathSciNet  Google Scholar 

  34. Whitney H.: Analytic extension of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36, 63–89 (1934)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

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. Department of Mathematics, Princeton University, Fine Hall Washington Road, Princeton, NJ, 08544, USA

    Charles Fefferman

  2. Department of Mathematics, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Pavel Shvartsman

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  1. Charles Fefferman
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  2. Pavel Shvartsman
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Correspondence to Pavel Shvartsman.

Additional information

This research was supported by Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF). Charles Fefferman was also supported in part by NSF Grant DMS- 1265524 and AFOSR Grant FA9550-12-1-0425.

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Fefferman, C., Shvartsman, P. Sharp Finiteness Principles For Lipschitz Selections. Geom. Funct. Anal. 28, 1641–1705 (2018). https://doi.org/10.1007/s00039-018-0467-6

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