On Bárány's theorems of Carathéodory and Helly type

Ehrhard Behrends

Studia Mathematica (2000)

  • Volume: 141, Issue: 3, page 235-250
  • ISSN: 0039-3223

Abstract

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The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows: if C n , n=1,2,..., are families of closed convex sets in a bounded subset of a separable Banach space X such that there exists a positive ε 0 with ⋂ C ∈ C n ( C ) ε = ∅ for ε < ε 0 , then there are C n ∈ C n with ⋂ n ( C n ) ε = ∅ for all ε < ε 0 ; here ( C ) ε denotes the collection of all x with distance at most ε to C.

How to cite

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Behrends, Ehrhard. "On Bárány's theorems of Carathéodory and Helly type." Studia Mathematica 141.3 (2000): 235-250. <http://eudml.org/doc/216782>.

@article{Behrends2000,
abstract = {The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows: if $C_\{n\}$, n=1,2,..., are families of closed convex sets in a bounded subset of a separable Banach space X such that there exists a positive $ε_\{0\}$ with $⋂_\{C ∈ C_\{n\}\} (C)_\{ε\} = ∅$ for $ε < ε_\{0\}$, then there are $C_\{n\} ∈ C_\{n\}$ with $⋂_\{n\} (C_\{n\})_\{ε\} = ∅$ for all $ε < ε_\{0\}$; here $(C)_\{ε\}$ denotes the collection of all x with distance at most ε to C.},
author = {Behrends, Ehrhard},
journal = {Studia Mathematica},
keywords = {Krein-Milman theorem; Helly; Helly-type theorem; Bárány; Carathéodory; RNP; Carathéodory theorem; Bárány’s theorems; Helly-Bárány theorem},
language = {eng},
number = {3},
pages = {235-250},
title = {On Bárány's theorems of Carathéodory and Helly type},
url = {http://eudml.org/doc/216782},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Behrends, Ehrhard
TI - On Bárány's theorems of Carathéodory and Helly type
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 3
SP - 235
EP - 250
AB - The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows: if $C_{n}$, n=1,2,..., are families of closed convex sets in a bounded subset of a separable Banach space X such that there exists a positive $ε_{0}$ with $⋂_{C ∈ C_{n}} (C)_{ε} = ∅$ for $ε < ε_{0}$, then there are $C_{n} ∈ C_{n}$ with $⋂_{n} (C_{n})_{ε} = ∅$ for all $ε < ε_{0}$; here $(C)_{ε}$ denotes the collection of all x with distance at most ε to C.
LA - eng
KW - Krein-Milman theorem; Helly; Helly-type theorem; Bárány; Carathéodory; RNP; Carathéodory theorem; Bárány’s theorems; Helly-Bárány theorem
UR - http://eudml.org/doc/216782
ER -

References

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  2. [2] M. Balaj and K. Nikodem, Remarks on Bárány's theorem and affine selections, preprint. 
  3. [3] I. Bárány, A generalization of Carathéodory's theorem, Discrete Math. 40 (1982), 141-152. Zbl0492.52005
  4. [4] I. Bárány, Carathéodory's theorem, colourful and applicable, in: Bolyai Soc. Math. Stud. 6, János Bolyai Math. Soc., 1997, 11-21. Zbl0883.52004
  5. [5] E. Behrends and K. Nikodem, A selection theorem of Helly type and its applications, Studia Math. 116 (1995), 43-48. Zbl0847.52004
  6. [6] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math. 64, Longman Sci. Tech., 1993. Zbl0782.46019
  7. [7] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977. 
  8. [8] J. Eckhoff, Helly, Radon, and Carathéodory type theorems, in: Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.), Elsevier, 1993, 389-448. Zbl0791.52009
  9. [9] R. C. James, A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. 80 (1974), 738-743. Zbl0286.46018
  10. [10] F. W. Levi, Eine Ergänzung zum Hellyschen Satze, Arch. Math. (Basel) 4 (1953), 222-224. Zbl0051.13701
  11. [11] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964). Zbl0141.12001
  12. [12] F. A. Valentine, Convex Sets, McGraw-Hill, 1964; reprinted by R. E. Krieger, 1976. Zbl0129.37203

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