Preface ix

set is nice and behaves predictably”. For the most part, the only properties of

the integral that the book uses are linearity (the integral of a linear combination

of two functions is the linear combination of the integrals of the functions) and

monotonicity (the integral of a non-negative function is non-negative). The relative

abundance of integrals in a textbook on convexity is explained by the fact that the

most natural way to define a linear functional is by using an integral of some sort. A

few exercises openly require some additional skills (knowledge of functional analysis

or representation theory).

When it comes to applications (often called “Propositions”), the reader is ex-

pected to have some knowledge in the general area which concerns the application.

Style. The numbering in each chapter is consecutive: for example, Theorem 2.1

is followed by Definition 2.2 which is followed by Theorem 2.3. When a reference

is made to another chapter, a roman numeral is included: for example, if Theorem

2.1 of Chapter I is referenced in Chapter III, it will be referred to as Theorem

I.2.1. Definitions, theorems and other numbered objects in the text (except figures)

are usually followed by a set of problems (exercises). For example, Problem 5

following Definition 2.6 in Chapter II will be referred to as Problem 5 of Section

2.6 from within Chapter II and as Problem 5 of Section II.2.6 from everywhere else

in the book. Figures are numbered consecutively throughout the book. There is a

certain difference between “Theorems” and “Propositions”. Theorems state some

general and fundamental convex properties or, in some cases, are called “Theorems”

historically. Propositions describe properties of particular convex sets or refer to

an application.

Problems. There are three kinds of problems in the text. The problems marked

by ∗ are deemed diﬃcult (they may be so marked simply because the author is

unaware of an easy solution). Problems with straightforward solutions are marked

by ◦. Solving a problem marked by ◦ is essential for understanding the material and

its result may be used in the future. Some problems are not marked at all. There are

no solutions at the end of the book and there is no accompanying solution manual

(that I am aware of) , which, in my opinion, makes the book rather convenient for

use in courses where grades are given. On the other hand, many of the diﬃcult

and some of the easy problems used later in the text are supplied with a hint to a

solution.

Acknowledgment. My greatest intellectual debt is to my teacher A.M. Vershik.

As a student, I took his courses on convexity and linear programming. Later, we

discussed various topics in convex analysis and geometry and he shared his notes

on the subject with me. We planned to write a book on convexity together and

actually started to write one (in Russian), but the project was effectively terminated

by my relocation to the United States. The overall plan, structure and scope of

the book have changed since then, although much has remained the same. All

unfortunate choices, mistakes, typos, blunders and other slips in the text are my

own. A.M. Vershik always insisted on a “dimension-free”approach to convexity,

whenever possible, which simplifies and makes transparent many fundamental facts,

and on stressing the idea of duality in the broadest sense. In particular, I learned

the algebraic approach to the Hahn-Banach Theorem (Sections II.1, III.1–3) and