On a theorem of Helly

We consider a group of problems related to the well-known Helly theorem on the intersections of convex bodies. We introduce convex subsetsK(?) of a compact convex setK defined by the relation $$K(f) = coleft{ {frac{N}{{N + 1}}x + frac{N}{{N + 1}}f(x)} right}{text{ }}(x in K subset mathbb{R}^N ),$$ where?: K→K are continuous mappings, and prove that the intersection ∩ _?∈F K(?) is not empty; hereF is the set of all continuous mappings?: K→K.