viii Preface

although not independent, provide enough material for several one-semester three-

credit courses.

One possibility is to follow discrete and combinatorial aspects of convexity:

combinatorial properties of convex sets (Chapter I) – the structure of some in-

teresting polytopes and polyhedra (the first part of Chapter II, some results of

Chapter IV and Chapter VI) – lattice points and convex bodies (Chapter VII) –

lattice points and polyhedra (Chapter VIII).

Another possibility is to follow the analytic line: basic properties of convex

sets (Chapter I) – the structure of some interesting non-polyhedral convex sets,

such as the moment cone, the cone of non-negative polynomials and the cone of

positive semidefinite matrices (Chapter II and some results of Chapter IV) – metric

properties of convex bodies (Chapter V).

Yet another possibility is to follow infinite-dimensional and dimension-free ap-

plications of convexity: basic properties of convex sets in a vector space (Chapter

I) – separation theorems and the structure of some interesting infinite-dimensional

convex sets (Chapter III) – linear inequalities and linear programming in an ab-

stract setting (Chapter IV).

The main focus of the book is on applications of convexity rather than on study-

ing convexity for its own sake. Consequently, mathematical applications range from

analysis and probability to algebra to combinatorics to number theory. Finite- and

infinite-dimensional optimization problems, such as the Transportation Problem,

the Diet Problem, problems of optimal control, statistics and approximation are

discussed as well.

The choice of topics covered in the book is entirely subjective. It is probably

impossible to write a textbook that covers “all” convexity just as it is impossible

to write a textbook that covers all mathematics. I don’t even presume to claim to

cover all “essential” or “important” aspects of convexity, although I believe that

many of the topics discussed in the book belong to both categories.

The audience. The book is intended for graduate students in mathematics and

other fields such as operations research, electrical engineering and computer science.

That was the typical audience for the courses that I taught. This is, of course,

reflected in the selection of topics covered in the book. Also, a significant portion

of the material is suitable for undergraduates.

Prerequisites. The main prerequisite is linear algebra, especially the coordinate-

free linear algebra. Knowledge of basic linear algebra should be suﬃcient for un-

derstanding the main convexity results (called “Theorems”) and solving problems

which address convex properties per se.

In many places, knowledge of some basic analysis and topology is needed. In

most cases, some general understanding coupled with basic computational skills

will be suﬃcient. For example, when it comes to the topology of Euclidean space,

it suﬃces to know that a set in Euclidean space is compact if and only if it is closed

and bounded and that a linear functional attains its maximum and minimum on

such a set. Whenever the book says “Lebesgue integral” or “Borel set”, it does so

for the sake of brevity and means, roughly, “the integral makes sense” and “the