Preprint No.
A-99-17
Ehrhard Behrends
On Barany's theorems of Caratheodory and Helly type
Abstract: The paper begins with a
self-contained and short development of \Ba's theorems of \Ca\ 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 \Ca-\Ba\ theorem has a counterpart in
arbitrary dimensions under suitable uniform compactness or uniform
boundedness conditions. The proper generalization of the Helly-\Ba\ theorem
reads as follows: if ${\cal C}_{n},n=1,\l$ are families of
closed convex sets in a bounded subset of a
separable Banach space $X$ such that there exists
a positive ${\eps}_{0}$ with $\bigcap_{C\in{\cal
C}_{n}}(C)_{{\eps}}=\emptyset$ for $\eps<\eps_{0}$, then there are $C_{n}\in{\ca
l C}_{n}$
with $\bigcap_{n}(C_{n})_{\eps}=\emptyset$ for all $\eps<{\eps}_{0}$; here $(C)_
{\eps}$ denotes
the collection of all $x$ with distance at most $\eps$ to $C$.
Keywords: Barany, Caratheodory, Helly, Helly-type theorem, Krein-Milman theorem, RNP
Mathematics Subject Classification (MSC91): 46B20 Geometry and structure of normed linear spaces
, 46B22 Radon-Nikodym, Kreuin-Milman and related properties, See also {46G10}
, 52A20 Convex sets in $n$ dimensions (including convex hypersurfaces), See also {53A07, 53C45}
, 52A35 Helly-type theorems
Language: ENG
Available: Pr-A-99-17.ps
Contact: Behrends, Ehrhard; Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (behrends@math.fu-berlin.de)
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