Preface

Convexity is very easy to define, to visualize and to get an intuition about. A set is

called convex if for every two points a and b in the set, the straight line interval [a, b]

is also in the set. Thus the main building block of convexity theory is a straight

line interval.

a

b

Convexity is more intuitive than, say, linear algebra. In linear algebra, the

interval is replaced by the whole straight line. We have some diﬃculty visualizing

a straight line because it runs unchecked in both directions.

On the other hand, the structure of convexity is richer than that of linear

algebra. It is already evident in the fact that all points on the line are alike whereas

the interval has two points, a and b, which clearly stand out.

Indeed, convexity has an immensely rich structure and numerous applica-

tions. On the other hand, almost every “convex” idea can be explained by a

two-dimensional picture. There must be some reason for that apart from the tau-

tological one that all our pictures are two-dimensional. One possible explanation is

that since the definition of a convex set involves only three points (the two points a

and b and a typical point x of the interval) and every three points lie in some plane,

whenever we invoke a convexity argument in our reasoning, it can be properly pic-

tured (moreover, since our three points a, b and x lie on the same line, we have

room for a fourth point which often plays the role of the origin). Simplicity, intu-

itive appeal and universality of applications make teaching convexity (and writing

a book on convexity) a rather gratifying experience.

About this book. This book grew out of sets of lecture notes for graduate courses

that I taught at the University of Michigan in Ann Arbor since 1994. Conse-

quently, this is a graduate textbook. The textbook covers several directions, which,

vii