On Bárány's theorems of Carathéodory and Helly type

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 ${\displaystyle C_n}$, 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 ${\displaystyle {\epsilon }_0}$ with ${\displaystyle {\bigcap }_{C\in C_n}{\left(C\right)}_{\epsilon }=\varnothing }$ for ${\displaystyle \epsilon <{\epsilon }_0}$, then there are ${\displaystyle C_n\in C_n}$ with ${\displaystyle {\bigcap }_n{\left(C_n\right)}_{\epsilon }=\varnothing }$ for all ${\displaystyle \epsilon <{\epsilon }_0}$; here ${\displaystyle {\left(C\right)}_{\epsilon }}$ denotes the collection of all x with distance at most ε to C.