Chapter 6 - SOME SPECIAL PROBLEMS

Summary

In this chapter we consider a number of particular problems which are either of importance in themselves or which illustrate the techniques available in this branch of mathematics. The problems are extremal geometric problems; that is to say, they are inequalities stated in terms of geometrical concepts. In any particular problem it is important to define the subclass of convex sets for which the inequality becomes an equality. The problems are of a type that can, in theory at any rate, be solved by the methods of the calculus of variations. In practice these methods are difficult to apply and cumbersome to handle. In the type of problem considered here the methods given are both more elegant and more precise than those of the calculus of variations. It is possible to give only a small selection of special problems, and the actual choice may strike the reader as rather arbitrary. It is arbitrary, since to give a systematic account of a representative selection of special problems would necessitate devoting more attention to them than would be proper in an introduction to the subject.

The success of this type of method depends to a great extent upon the simplicity of the structure of the extremal figures. Where the extremal figure is unique (to within a congruence or an affine transformation say), then the method is likely to be applicable. Where the extremal figures are many and cannot easily be described in geometrical language, there the method will be difficult to apply or even impossible.