An extension of Radon's theorem
John R. Reay · Illinois Journal of Mathematics · 1968
introductionLet "m-set" mean a set of m points in the d-dimensional space R .An m-set is said to be (r, )-divisible if it can be partitioned into r pair-wise dis- joint subsets in such a way that the intersection of the convex hulls of these r subsets is at least It-dimensional.(We always assume 0 _ / _ d.The empty set is (-1)-dimensional, while 0-dimensional sets are non-empty.)A classic theorem of J. Radon [5] asserts that each (d 2)-set is (2, 0)- divisible.The first generalization for r > 2 was given by R. Rado [4].B. Birch [1] conjectured (and proved for d 2) that each ((d -1)(r 1) W 1)-set is (r, 0)-divisible, while H. Tverberg [6] estab- lished this coniecture for all values of d.It is clear that if/ > 0 then other conditions on a given m-set S besides a lower bound on its cardinality are necessary if S is to be (r, /)-divisible.For example, if all the points of S were on a line in R , no subset would have a convex hull of dimension greater than one.The purpose of this paper is to consider various types of independ- ence that may be imposed upon an m-set to insure r, /c -divisibility (Section 3) and to prove the following theorem which extends the results mentioned