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 ofR2to be the restriction of a function from the Sobolev space W∞2(R2).A similar result is proved for the space of functions onR2satisfying 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
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Asplund, E. Comparison between plane symmetric convex bodies and parallelograms,Math. Scand.,8, 171–180, (1960).MathSciNetMATHGoogle Scholar
Aubin, J.-P. and Cellina, A.Differential Inclusions, Springer-Verlag, Berlin, (1984).MATHGoogle Scholar
Aubin, J.-P. and Frankowska, H.Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser, Boston, (1990).MATHGoogle Scholar
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
Boltyanski, V., Martini, H., and Soltan, P.S.Excursions into Combinatorial Geometry, Universitext. Springer-Verlag, Berlin, xiv+418, (1997).MATHGoogle Scholar
Brudnyi, Yu. and Shvartsman, P. Generalizations of Whitney’s extension theorem,Intern. Math. Research Notices,3, 129–139, (1994).CrossRefMathSciNetGoogle Scholar
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
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
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
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
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
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
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
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
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