Geometry of Convex Sets

Many properties of convex sets can be discovered using linear structures; however, the concepts of a metric space, i.e. open sets, closed sets, bounded sets, compact sets, etc., need to be incorporated in order to prove more valuable results. While convexity can be defined using only the linear structure, the authors blend linear and topological concepts to highlight the practicality of the results. In the first chapter, linear or vector space notions of addition and scalar multiplication are introduced in addition to linear subspaces, linear functionals, and hyperplanes. The second chapter presents the affine concepts of flats and lines by studying the incidence of points and lines. The authors provide a proof of Sylvester's theorem and a brief introduction to finite, countable, and uncountable sets. The third chapter discusses the notion of topology in the setting of metrics derived from a norm on the n-dimensional space. The basic properties of compact sets are provided as well as geometric examples to illustrate that all these notions can be discussed in terms of convergence of sequences and subsequences within the setting of a metric space. In the fourth chapter, the notion of convexity and the basic properties of convex sets are discussed. The authors define the convex hull of a set in addition to the interior and closure of convex sets. The fifth and last chapter focuses on Helly's theorem and related applications that involve transversals of families of pairwise disjoint compact convex subsets of the plane. Lastly, a proof of Borsuk's problem in the plane using Pal's theorem and also Melzak's proof of Borsuk's problem for smooth sets of constant width 1 in Rn is presented.