On Bárány's theorems of Carathéodory and Helly type (Q2717546)

The author gives a self-contained exposition of Bárány's theorems of Carathéodory and Helly type in finite-dimensional spaces, with some new variants. Infinite-dimensional variants of both results are then investigated, under proper compactness assumptions. The generalized Helly-Bárány theorem states:NEWLINENEWLINENEWLINEif \(({\mathcal C}_n)_{n\in\mathbb{N}}\) are families of closed convex sets in a bounded subset of a separable Banach space \(X\) such that for some \(\varepsilon_0> 0\bigcap_{C\in{\mathcal C}_n}(C)_\varepsilon= \emptyset\) for \(\varepsilon< \varepsilon_0\), then there are \(C_n\in{\mathcal C}_n\) with \(\bigcap_n (C_n)_\varepsilon=\emptyset\) for \(\varepsilon< \varepsilon_0\). Here \((C)_\varepsilon\) denotes the se of all \(x\) with distance at most \(\varepsilon\) to \(C\).