Advertisement

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

  • Pavel Shvartsman
Article

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.

Math Subject Classifications

54C60 54C65 46E35 52A35 

Key Words and Phrases

set-valued mapping Lipschitz selection Whitney’s extension problem traces of smooth functions Helly-type theorems 

References

  1. [1]
    Artstein, Z. Extension of Lipschitz selections and an application to differential inclusions,Nonlinear Anal.,16, 701–704, (1991).CrossRefMathSciNetMATHGoogle Scholar
  2. [2]
    Asplund, E. Comparison between plane symmetric convex bodies and parallelograms,Math. Scand.,8, 171–180, (1960).MathSciNetMATHGoogle Scholar
  3. [3]
    Aubin, J.-P. and Cellina, A.Differential Inclusions, Springer-Verlag, Berlin, (1984).MATHGoogle Scholar
  4. [4]
    Aubin, J.-P. and Frankowska, H.Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser, Boston, (1990).MATHGoogle Scholar
  5. [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. [6]
    Boltyanski, V., Martini, H., and Soltan, P.S.Excursions into Combinatorial Geometry, Universitext. Springer-Verlag, Berlin, xiv+418, (1997).MATHGoogle Scholar
  7. [7]
    Brudnyi, Yu. and Shvartsman, P. Generalizations of Whitney’s extension theorem,Intern. Math. Research Notices,3, 129–139, (1994).CrossRefMathSciNetGoogle Scholar
  8. [8]
    Brudnyi, Yu. and Shvartsman, P. The Whitney problem of existence of a linear extension operator,J. Geom. Anal.,7(4), 515–574, (1997).MathSciNetMATHGoogle Scholar
  9. [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).CrossRefMathSciNetMATHGoogle Scholar
  10. [10]
    Brudnyi, Yu. and Shvartsman, P. Whitney’s extension problem for multivariate C1,ω-functions,Trans. Am. Math. Soc.,353, 2487–2512, (2001).CrossRefMathSciNetMATHGoogle Scholar
  11. [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).CrossRefMathSciNetGoogle Scholar
  12. [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. [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).CrossRefMathSciNetMATHGoogle Scholar
  14. [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).CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    Glaeser, G. Étude de quelques algebres Tayloriennes,J. d’Analyse Math.,6, 1–125, (1958).MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Grünbaum, B.Convex Polytopes, John Wiley & Sons, New York, (1967).MATHGoogle Scholar
  17. [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).CrossRefMathSciNetMATHGoogle Scholar
  18. [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).MathSciNetGoogle Scholar
  19. [19]
    Mattioli, J. Minkowski operations and vector spaces,Set-Valued Anal.,3(1), 33–50, (1995).CrossRefMathSciNetMATHGoogle Scholar
  20. [20]
    Posicelskii, E.D. Lipschitzian mappings in the space of convex bodies,Optimizacija,4(21), 83–89, (1971), (Russian).MathSciNetGoogle Scholar
  21. [21]
    Posicelskii, E.D. The characterization of the Steiner point,Mat. Zametki,14, 243–247, (1973), (Russian).MathSciNetGoogle Scholar
  22. [22]
    Przesławski, K. and Yost, D. Continuity properties of selectors and Michael’s theorem,Mich. Math. J.,36, 113–134, (1989).CrossRefMATHGoogle Scholar
  23. [23]
    Przesławski, K. and Yost, D. Lipschitz retracts, selectors and extensions,Mich. Math. J.,42, 555–571, (1995).CrossRefMATHGoogle Scholar
  24. [24]
    Przesławski, K. Centres of convex sets inL p metrics,J. Approx. Theory,85, 288–296, (1996).CrossRefMathSciNetMATHGoogle Scholar
  25. [25]
    Przesławski, K. Lipschitz continuous selectors, I, linear selectors,J. Convex Anal.,5(2), 249–267, (1998).MathSciNetMATHGoogle Scholar
  26. [26]
    Repovš, D. and Semenov, P.V.Continuous Selections of Multivalued Mappings, Mathematics and its Applications, 455. Kluwer Academic Publishers, Dordrecht, viii+356, (1998).MATHGoogle Scholar
  27. [27]
    Saint-Pierre, J. Point de Steiner et sections lipschitziennes, inSém. Anal. Convexe,15(7), pp. 42, (1985).MathSciNetGoogle Scholar
  28. [28]
    Schneider, R. On Steiner points of convex bodies,Israel J. Math.,9, 241–249, (1971).CrossRefMathSciNetMATHGoogle Scholar
  29. [29]
    Shephard, G.C. The Steiner point of a convex polytope,Canad. J. Math.,18, 1294–1300, (1966).MathSciNetMATHGoogle Scholar
  30. [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).MathSciNetGoogle Scholar
  31. [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).MathSciNetMATHGoogle Scholar
  32. [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).MathSciNetGoogle Scholar
  33. [33]
    Shvartsman, P. On Lipschitz selections of affine-set valued mappings,Geom. Funct. Anal. (GAFA),11(4), 840–868, (2001).CrossRefMathSciNetMATHGoogle Scholar
  34. [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).Google Scholar
  35. [35]
    Steiner, J.Gesammelte Werke, 2 vols., Berlin, (1881, 1882).Google Scholar
  36. [36]
    Ubhaya, V.A. Lipschitzian selections in best approximation by continuous functions,J. Approx. Theory,61(1), 40–52, (1990).CrossRefMathSciNetMATHGoogle Scholar
  37. [37]
    Whitney, H. Analytic extension of differentiable functions defined in closed sets,Trans. Am. Math. Soc.,36, 63–89, (1934).CrossRefMathSciNetGoogle Scholar
  38. [38]
    Whitney, H. Differentiable functions defined in closed sets,I, Trans. Am. Math. Soc. 36, 369–387, (1934).CrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2002

Authors and Affiliations

  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

Personalised recommendations