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
(1.1) Euclidean space. The d-dimensional Euclidean space
consists of all d-
tuples x = (ξ1, . . . , ξd) of real numbers. We call an element of
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),
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