Radon's Theorem - Topological Radon Theorem

Topological Radon Theorem

A topological generalization of Radon's theorem states that, if ƒ is any continuous function from a (d + 1)-dimensional simplex to d-dimensional space, then the simplex has two disjoint faces whose images under ƒ are not disjoint. Radon's theorem itself can be interpreted as the special case in which ƒ is the unique affine map that takes the vertices of the simplex to a given set of d + 2 points in d-dimensional space.

More generally, if K is any (d + 1)-dimensional compact convex set, and ƒ is any continuous function from K to d-dimensional space, then there exists a linear function g such that some point where g achieves its maximum value and some other point where g achieves its minimum value are mapped by ƒ to the same point. In the case where K is a simplex, the two simplex faces formed by the maximum and minimum points of g must then be two disjoint faces whose images have a nonempty intersection. This same general statement, when applied to a hypersphere instead of a simplex, gives the Borsuk–Ulam theorem, that ƒ must map two opposite points of the sphere to the same point.