Radon's Theorem - Related Concepts

Related Concepts

The Radon point of three points in a one-dimensional space is just their median. The geometric median of a set of points is the point minimizing the sum of distances to the points in the set; it generalizes the one-dimensional median and has been studied both from the point of view of facility location and robust statistics. For sets of four points in the plane, the geometric median coincides with the Radon point.

Another generalization for partition into r sets was given by Helge Tverberg (1966) and is now known as Tverberg's theorem. It states that for any set of

points in Euclidean d-space, there is a partition into r subsets whose convex hulls intersect in at least one common point.

Carathéodory's theorem states that any point in the convex hull of some set of points is also within the convex hull of a subset of at most d + 1 of the points; that is, that the given point is part of a Radon partition in which it is a singleton. One proof of Carathéodory's theorem uses a technique of examining solutions to systems of linear equations, similar to the proof of Radon's theorem, to eliminate one point at a time until at most d + 1 remain.

Concepts related to Radon's theorem have also been considered for convex geometries, families of finite sets with the properties that the intersection of any two sets in the family remains in the family, and that the empty set and the union of all the sets belongs to the family. In this more general context, the convex hull of a set S is the intersection of the family members that contain S, and the Radon number of a space is the smallest r such that any r points have two subsets whose convex hulls intersect. Similarly, one can define the Helly number h and the Carathéodory number c by analogy to their definitions for convex sets in Euclidean spaces, and it can be shown that these numbers satisfy the inequalities h < r ≤ ch + 1.

In an arbitrary undirected graph, one may define a convex set to be a set of vertices that includes every induced path connecting a pair of vertices in the set. With this definition, every set of ω + 1 vertices in the graph can be partitioned into two subsets whose convex hulls intersect, and ω + 1 is the minimum number for which this is possible, where ω is the clique number of the given graph. For related results involving shortest paths instead of induced paths see Chepoi (1986) and Bandelt & Pesch (1989).