Chapter 2 - HELLY'S THEOREM AND ITS APPLICATIONS

Summary

One of the most striking properties of Euclidean n-dimensional space is a result on the intersection of convex sets due to Helly. This property is closely related to Carathéodory's theorem on the convex cover of a given set, and the relationship is connected with duality. Carathéodory's theorem implies Helly's theorem, and conversely also Helly's theorem implies the dual of Carathéodory's. Here of course we are using the concept of duality in a descriptive and imprecise sense.

The properties of convex sets which were developed in Chapter 1 are true in one form or another in Banach spaces of either finite or infinite dimension. This is no longer the case with the theorems that are to be proved in the present chapter. A vector space which satisfies Helly's theorem is essentially one whose dimension is finite. It is possible to generalize Helly's theorem by a process of axiomatization, but we shall not do so here.

Radon's proof of Helly's theorem

We give here a simple analytical proof of Helly's theorem due to Radon.

Theorem 17. Helly's theorem. A finite class of N convex sets in Rⁿis such that N ≥ n + 1, and to every subclass which contains n + 1 members there corresponds a point of Rⁿwhich belongs to every member of the subclass. Under these conditions there is a point which belongs to every member of the given class of N convex sets.