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
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