CHAPTER 2.1 - Helly, Radon, and Carathéodory Type Theorems

Abstract

This chapter discusses applications and generalizations of the classical theorems of Helly, Radon, and Carathéodory, as well as their ramifications in the context of combinatorial convexity theory. These theorems stand at the origin of what is known today as the combinatorial geometry of convex sets. Helly's theorem states that: letting K be a family of convex sets in ℝd, and supposing K is finite or each member of K is compact; if every d + 1 or fewer members of K have a common point, then there is a point common to all members of K. Radon's theorem states that: letting X be a set of d + 2 or more points in ℝd; then X contains two disjoint subsets whose convex hulls have a common point. Carathéodory's theorem states that: letting X be a set in ℝd and p a point in the convex hull of X. Then there is a subset Y of X consisting of d + 1 or fewer points such that p lies in the convex hull of Y.