On Bárány's theorems of Carathéodory and Helly type

Ehrhard Behrends¹

¹

Studia Mathematica, Tome 141 (2000) no. 3, pp. 235-250

Résumé

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.

Détail

DOI : 10.4064/sm-141-3-235-250

Keywords: Krein-Milman theorem, Helly, Helly-type theorem, Bárány, Carathéodory, RNP