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

Article

DOI: 10.1007/BF02922044

Cite this article as:
Shvartsman, P. J Geom Anal (2002) 12: 289. doi:10.1007/BF02922044

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 ofR2to be the restriction of a function from the Sobolev space W2(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 

Copyright information

© Mathematica Josephina, Inc. 2002

Authors and Affiliations

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

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