On Bárány's theorems of Carathéodory and Helly type
Behrends, Ehrhard
Studia Mathematica, Tome 141 (2000), p. 235-250 / Harvested from The Polish Digital Mathematics Library
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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 Cn, 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 ε0 with ⋂C∈Cn(C)ε=∅ for ε<ε0, then there are Cn∈Cn with ⋂n(Cn)ε=∅ for all ε<ε0; here (C)ε denotes the collection of all x with distance at most ε to C.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:216782 
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     year = {2000},
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Behrends, Ehrhard. On Bárány's theorems of Carathéodory and Helly type. Studia Mathematica, Tome 141 (2000) pp. 235-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv141i3p235bwm/

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