Chapter I

Convex Sets at Large

We define convex sets and explore some of their fundamental properties. In this

chapter, we are interested in the “global” properties of convex sets as opposed to the

“local” properties studied in the next chapter. Namely, we are interested in what

a convex set looks like as a whole, how convex sets may intersect and how they be-

have with respect to linear transformations. In contrast, in the next chapter, we will

discuss what a convex set looks like in a neighborhood of a point. The landmark

results of this chapter are classical theorems of Carath´ eodory, Radon and Helly

and the geometric construction of the Euler characteristic. We apply our results

to study positive multivariate polynomials, the problem of uniform (Chebyshev)

approximation and some interesting valuations on convex sets, such as the intrin-

sic volumes. Exercises address some other applications (such as the Gauss-Lucas

Theorem), discuss various ramifications of the main results (such as the Fractional

Helly Theorem or the Colored Carath´ eodory Theorem) and preview some of the

results of the next chapters (such as the Brickman Theorem, the Schur-Horn Theo-

rem and the Birkhoff-von Neumann Theorem). We introduce two important classes

of convex sets, polytopes and polyhedra, discussed throughout the book.

1. Convex Sets. Main Definitions, Some Interesting

Examples and Problems

First, we set the stage where the action is taking place. Much of the action, though

definitely not all, happens in Euclidean space

Rd.

(1.1) Euclidean space. The d-dimensional Euclidean space

Rd

consists of all d-

tuples x = (ξ1, . . . , ξd) of real numbers. We call an element of

Rd

a vector or (more

often) a point. We can add points: we say that

z = x + y for x = (ξ1, . . . , ξd), y = (η1, . . . , ηd) and z = (ζ1, . . . , ζd),

1

http://dx.doi.org/10.1090/gsm/054/01