2 - Convexity

Summary

Introduction

Convex analysis is an indispensible tool of mathematical statistics. The purpose of this chapter is to provide a self-contained body of central results which are useful for decision theory in general and which are particularly useful for the topic covered in this book.

Several excellent textbooks are available in this area. Some well known books in decision theory, e.g. Blackwell & Girshick (1954), Ferguson (1967) and LeCam (1986), also contain quite an amount of convex analysis. Another highly relevant work is the book by Marshall & Olkin (1979).

References to original sources, with a few exceptions, are not given. They may be found either in the cited textbooks or in some of the other textbooks which are included in the references. The references are mainly books which have been particularly useful to the author.

The basic concepts and notions are introduced within the framework of a general real linear space, in section 2.1. However results which do not follow fairly directly from the definitions are usually only given in the finite dimensional case. An exception is the Hahn-Banach theorem. The spadework contained in the ‘standard’ proof of that theorem is finite dimensional. As we shall see, an inspection of this proof provides criteria for measurability of the guaranteed extension and this yields a substantial part of an important decomposition theorem in Strassen (1965). The complete proof of this theorem is provided in complement 22 in chapter 10.