# On Bárány's theorems of Carathéodory and Helly type

Studia Mathematica (2000)

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

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topBehrends, 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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- [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] F. W. Levi, Eine Ergänzung zum Hellyschen Satze, Arch. Math. (Basel) 4 (1953), 222-224. Zbl0051.13701
- [11] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964). Zbl0141.12001
- [12] F. A. Valentine, Convex Sets, McGraw-Hill, 1964; reprinted by R. E. Krieger, 1976. Zbl0129.37203

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