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Lipschitz selections of set-valued mappings and Helly’s theorem

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Abstract

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family.

Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2m+1 points, the restriction F¦M′ of F to M′ has a selection fM′ (i. e., fM′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒM′(x) − ƒM′(y)‖X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2m+1, is sharp.

If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R 2 to be the restriction of a function from the Sobolev space W 2∞ (R 2).A similar result is proved for the space of functions on R 2 satisfying the Zygmund condition.

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References

  1. Artstein, Z. Extension of Lipschitz selections and an application to differential inclusions,Nonlinear Anal.,16, 701–704, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  2. Asplund, E. Comparison between plane symmetric convex bodies and parallelograms,Math. Scand.,8, 171–180, (1960).

    MathSciNet  MATH  Google Scholar 

  3. Aubin, J.-P. and Cellina, A.Differential Inclusions, Springer-Verlag, Berlin, (1984).

    MATH  Google Scholar 

  4. Aubin, J.-P. and Frankowska, H.Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser, Boston, (1990).

    MATH  Google Scholar 

  5. Benyamini, Y. and Lindenstrauss, J. Geometric nonlinear functional analysis, Vol. 1,American Mathematical Society Colloquium Publications, 48, American Mathematical Society, Providence, RI, xii+488, (2000).

    Google Scholar 

  6. Boltyanski, V., Martini, H., and Soltan, P.S.Excursions into Combinatorial Geometry, Universitext. Springer-Verlag, Berlin, xiv+418, (1997).

    MATH  Google Scholar 

  7. Brudnyi, Yu. and Shvartsman, P. Generalizations of Whitney’s extension theorem,Intern. Math. Research Notices,3, 129–139, (1994).

    Article  MathSciNet  Google Scholar 

  8. Brudnyi, Yu. and Shvartsman, P. The Whitney problem of existence of a linear extension operator,J. Geom. Anal.,7(4), 515–574, (1997).

    MathSciNet  MATH  Google Scholar 

  9. Brudnyi, Yu. and Shvartsman, P. The trace of jet spaceΛ ω to an arbitrary closed subset of ℝn,Trans. Am. Math. Soc.,350, 1519–1553, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  10. Brudnyi, Yu. and Shvartsman, P. Whitney’s extension problem for multivariate C1,ω-functions,Trans. Am. Math. Soc.,353, 2487–2512, (2001).

    Article  MathSciNet  MATH  Google Scholar 

  11. Dal Maso, G., Goncharov, V.V., and Ornelas, A.A. Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient,Nonlinear Anal.,37(6), Ser. A: Theory Methods, 707–717, (1999).

    Article  MathSciNet  Google Scholar 

  12. Danzer, L., Grünbaum, B., and Klee, V. Helly’s theorem and its relatives,Am. Math. Soc. Symp. on Convexity, Seattle, Proc. Symp. Pure Math., vol.7, 101–180,Am. Math. Soc., Providence, RI, (1963).

    Google Scholar 

  13. Deutsch, F., Li, Wu., and Park, S.-Ho. Characterizations of continuous and Lipschitz continuous selections for metric projections in normed linear spaces,J. Approx. Theory,58, 297–314, (1989).

    Article  MathSciNet  MATH  Google Scholar 

  14. Deutsch, F. and Li, Wu. Strong uniqueness, Lipschitz continuity, and continuous selections for metric projections in L1,J. Approx. Theory,66(2), 198–224, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  15. Glaeser, G. Étude de quelques algebres Tayloriennes,J. d’Analyse Math.,6, 1–125, (1958).

    Article  MathSciNet  MATH  Google Scholar 

  16. Grünbaum, B.Convex Polytopes, John Wiley & Sons, New York, (1967).

    MATH  Google Scholar 

  17. Jonsson, A. The trace of the Zygmund class Λk(R) to closed sets and interpolating polynomials,J. Approx. Theory,44(1), 1–13, (1985).

    Article  MathSciNet  MATH  Google Scholar 

  18. Marinov, A.V. Lipschitz constants of the metric ∈-projection operator in spaces with given moduli of convexity and smoothness,Izv. Ross. Akad. Nauk Ser. Mat.,62(2), 103–130, (1998), (Russian); translation inIzv. Math., 62(2), 313–338, (1998).

    MathSciNet  Google Scholar 

  19. Mattioli, J. Minkowski operations and vector spaces,Set-Valued Anal.,3(1), 33–50, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  20. Posicelskii, E.D. Lipschitzian mappings in the space of convex bodies,Optimizacija,4(21), 83–89, (1971), (Russian).

    MathSciNet  Google Scholar 

  21. Posicelskii, E.D. The characterization of the Steiner point,Mat. Zametki,14, 243–247, (1973), (Russian).

    MathSciNet  Google Scholar 

  22. Przesławski, K. and Yost, D. Continuity properties of selectors and Michael’s theorem,Mich. Math. J.,36, 113–134, (1989).

    Article  MATH  Google Scholar 

  23. Przesławski, K. and Yost, D. Lipschitz retracts, selectors and extensions,Mich. Math. J.,42, 555–571, (1995).

    Article  MATH  Google Scholar 

  24. Przesławski, K. Centres of convex sets inL p metrics,J. Approx. Theory,85, 288–296, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  25. Przesławski, K. Lipschitz continuous selectors, I, linear selectors,J. Convex Anal.,5(2), 249–267, (1998).

    MathSciNet  MATH  Google Scholar 

  26. Repovš, D. and Semenov, P.V.Continuous Selections of Multivalued Mappings, Mathematics and its Applications, 455. Kluwer Academic Publishers, Dordrecht, viii+356, (1998).

    MATH  Google Scholar 

  27. Saint-Pierre, J. Point de Steiner et sections lipschitziennes, inSém. Anal. Convexe,15(7), pp. 42, (1985).

    MathSciNet  Google Scholar 

  28. Schneider, R. On Steiner points of convex bodies,Israel J. Math.,9, 241–249, (1971).

    Article  MathSciNet  MATH  Google Scholar 

  29. Shephard, G.C. The Steiner point of a convex polytope,Canad. J. Math.,18, 1294–1300, (1966).

    MathSciNet  MATH  Google Scholar 

  30. Shvartsman, P.A. Lipschitz selections of multivalued mappings and the traces of the Zygmund class functions to an arbitrary compact,Dokl. Akad. Nauk SSSR,276(3), 559–562, (1984); English transl. inSoviet. Math. Dokl., 29(3), 565–568, (1984).

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  32. Shvartsman, P.A. K-functionals of weighted Lipschitz spaces and Lipschitz selections of multivalued mappings,Interpolation Spaces and Related Topics, Israel Math. Conf. Proc.,5, 245–268, (1992).

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Skaletskii, A.G. Uniformly continuous selections in Fréchet spaces,Vestn. Mosk. Gos. Univ. Ser. I. Mat. Mekh., (2), 24–28, (1985), (Russian); Engl. transl. inMoscow Univ. Math. Bull.,40(2), 29–53, (1985).

  35. Steiner, J.Gesammelte Werke, 2 vols., Berlin, (1881, 1882).

  36. Ubhaya, V.A. Lipschitzian selections in best approximation by continuous functions,J. Approx. Theory,61(1), 40–52, (1990).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  38. Whitney, H. Differentiable functions defined in closed sets,I, Trans. Am. Math. Soc. 36, 369–387, (1934).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Pavel Shvartsman

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  1. Pavel Shvartsman
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Correspondence to Pavel Shvartsman.

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Communicated by Yoram Sagher

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Shvartsman, P. Lipschitz selections of set-valued mappings and Helly’s theorem. J Geom Anal 12, 289–324 (2002). https://doi.org/10.1007/BF02922044

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  • DOI: https://doi.org/10.1007/BF02922044

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