1 - Why convex?

Summary

The first modern formalization of the concept of convex function appears in J. L. W. V. Jensen, “Om konvexe funktioner og uligheder mellem midelvaerdier.” Nyt Tidsskr. Math. B 16(1905), pp. 49–69. Since then, at first referring to “Jensen's convex functions,” then more openly, without needing any explicit reference, the definition of convex function becomes a standard element in calculus handbooks.

(A. Guerraggio and E. Molho)

Convexity theory … reaches out in all directions with useful vigor. Why is this so? Surely any answer must take account of the tremendous impetus the subject has received from outside of mathematics, from such diverse fields as economics, agriculture, military planning, and flows in networks. With the invention of high-speed computers, large-scale problems from these fields became at least potentially solvable. Whole new areas of mathematics (game theory, linear and nonlinear programming, control theory) aimed at solving these problems appeared almost overnight. And in each of them, convexity theory turned out to be at the core. The result has been a tremendous spurt in interest in convexity theory and a host of new results.

(A.Wayne Roberts and Dale E. Varberg)

Why ‘convex’?

This introductory polemic makes the case for a study focusing on convex functions and their structural properties. We highlight the centrality of convexity and give a selection of salient examples and applications; many will be revisited in more detail later in the text – and many other examples are salted among later chapters.