On a theorem of Helly

Abstract

We consider a group of problems related to the well-known Helly theorem on the intersections of convex bodies. We introduce convex subsetsK(ƒ) of a compact convex setK defined by the relation

$$K(f) = co\left\{ {\frac{N}{{N + 1}}x + \frac{N}{{N + 1}}f(x)} \right\}{\text{ }}(x \in K \subset \mathbb{R}^N ),$$

whereƒ: K→K are continuous mappings, and prove that the intersection ∩ _ƒ∈F K(ƒ) is not empty; hereF is the set of all continuous mappingsƒ: K→K.