Convexity and Extremal Principles

Abstract

In the preceding chapter we showed how existence propositions for extremal problems are obtained with the aid of compactness arguments. A second basic strategy for obtaining existence propositions consists in considering convexity instead of compactness. Figure 39.1 shows the logical connections. We place the Hahn-Banach theorem at the pinnacle; in the final analysis this theorem goes back to the central fixed point theorem of Bourbaki and Kneser, in Chapter 11, via Zorn’s lemma. The separation theorems for convex sets and the Krein extension theorem for positive functionals follow from the Hahn-Banach theorem. These three theorems are standard results of functional analysis. We summarize them in Section 39.1 without proofs. The proofs can be found, e.g., in Edwards (1965, M). In fact these three theorems, which are framed in Fig. 39.1, are mutually equivalent if they are appropriately formulated. They represent different conceptions of a general fundamental principle of geometric functional analysis, which finds its most suggestive geometrical form in the separation theorems for convex sets. These equivalences are discussed in Holmes (1975, M), page 95 (cf. Problem 39.13).

It seems to me that the notion of convex function is just as fundamental as positive or increasing function. If I am not mistaken in this, the notion ought to find its place in elementary expositions of the theory of real functions.

J. L. W. V. Jensen, 1906

The study of convex sets is a branch of geometry, analysis, and linear algebra that has numerous connections with other areas of mathematics and serves to unify many apparently diverse mathematical phenomena.

Victor Klee, 1950