2 - Convex bodies

Summary

As was stated in the introduction to Chapter 0, it is difficult to discuss the theory of convex sets without using such terms as closed, bounded and interior point. It is clear from Theorem 1.2.3 that, for a finite dimensional space X, these terms refer to the unique Hausdorff linear topology on X. One may think of this topology as coming from a Euclidean structure on X or from an arbitrary norm on X.

In the subsequent chapters there will be occasion to use a wide assortment of material from the theory of convex bodies. While it is not assumed that the reader is familiar with this material, to include a complete discussion of all the results that we shall need (especially the various inequalities relating to volumes and mixed volumes) would require a separate book. In this chapter we shall therefore only summarize the material, stating the main results, giving outlines for the proofs of some of them and indicating where complete proofs can be found. A full account of the material is contained in several places. The first systematic account is in Bonnesen and Fenchel, of which there is a relatively recent English translation [51]. The book of Eggleston [138] contains most of what we need. The best and most up-to-date reference is the book by Schneider [479], which also contains a wealth of references to the original sources.