Convex Sets: Basic Properties

Abstract

• Why. The central object of convex analysis is a convex set. Whenever weighted averages play a role, such as in the analysis by Nash of the question ‘what is a fair bargain?’, one is led to consider convex sets.

• What. An introduction to convex sets and to the reduction of these to convex cones (‘homogenization’) is given. The ray, hemisphere and top-view models to visualize convex sets are explained. The classic results of Radon, Helly and Carathéodory are proved using homogenization. It is explained how convex cones are equivalent to preference relations on vector spaces. It is proved that the Minkowski sum of a large collection of sets of vectors approximates the convex hull of the union of the collection (lemma of Shapley–Folkman).

Road Map For each figure, the explanation below the figure has to be considered as well.

• Definition 1.2.1 and Fig. 1.3 (convex set).

• Definition 1.2.6 and Fig. 1.4 (convex cone; note that a convex cone is not required to contain the origin).

• Figure 1.5 (the ray, hemisphere and top-view model for a convex cone in dimension one; similar in higher dimensions: for dimension two, see Figs. 1.8 and 1.9).

• Figure 1.10 (the (minimal) homogenization of a convex set in dimension two and in the bounded case).

• Figure 1.12 (the ray, hemisphere and top-view model for a convex set in dimension one; for dimension two, see Figs. 1.13 and 1.14).

• Definition 1.7.2 and Fig. 1.15 (convex hull).

• Definition 1.7.4 and Fig. 1.16 (conic hull; note that the conic hull contains the origin).

• Definitions 1.7.6, 1.7.9, Proposition 1.7.8, its proof and the explanation preceding it (the simplest illustration of the homogenization method).

• Section 1.8.1 (the ingredients for the construction of binary operations for convex sets). • Theorems 1.9.1, 1.10.1, 1.12.1 and the structure of their proofs (theorems of Radon, Helly, Carathéodory, proved by the homogenization method).

• Definitions 1.12.6, 1.12.9 and 1.12.11 (polyhedral set, polyhedral cone and Weyl’s theorem).

• Definition 1.10.5, Corollary 1.12.4 and its proof (use of compactness in the proof).

• The idea of Sect. 1.13 (preference relations: a different angle on convex cones).