On Barany's theorems of Caratheodory and Helly type

The paper begins with a self-contained and short development of Barany's theorems of Caratheodory 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 Caratheodory-Barany theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Barany 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 EO with boolean AND (C is an element of Cn)(C)(epsilon) = 0 for epsilon < <epsilon>(0), then there are C-n is an element of C-n with boolean AND (n)(C-n)(epsilon) = 0 for all epsilon < <epsilon>(0); here (C)(epsilon) denotes the collection of all x with distance at most epsilon to C.