Chapter 4 - APPROXIMATIONS TO CONVEX SETS. THE BLASCHKE SELECTION THEOREM

Summary

It is essential in many extremal geometrical problems to be able to say whether or not a particular configuration can actually occur. If one knows that the configuration can be attained the problem is usually simplified. This is because one is interested in the extremal values of some functions of the configuration, and those configurations for which this extremal is attained form a class whose structure is essentially simpler than that of the original class. This simplification is particularly striking if the extremal configuration is unique.

The principal object of this chapter is to prove Blaschke's selection theorem. This theorem, which asserts that the class of closed convex subsets of a closed bounded convex set of Rⁿ can be made into a compact metric space, enables one to assert the existence of extremal configurations in many cases. The practical importance of this theorem cannot be overemphasized, and some examples of its use will be given in Chapters 6 and 7.

In this chapter and in all succeeding chapters we shall use the phrase ‘convex set’ to mean ‘closed bounded convex set’, and ‘convex body’ to mean ‘closed bounded convex set with interior points’. Where a particular convex set is not closed or bounded we shall say so explicitly and we shall sometimes use the phrase ‘closed bounded convex set’ where it is desirable to emphasize the closed bounded nature of the set.