On the Geometry and Computational Complexity of Radon Partitions in the Integer Lattice

The following integer analogue of a Radon partition in affine space rd is studied: A partition (S, T) of a set of integer points in ,]d is an integral Radon partition if the convex hulls of S and T have an integer point in common. The Radon number r(d) of an appropriate convexity space on the integer lattice Zd is then the infimum over those natural numbers n such that any set of n points or more in Zd has an integral Radon partition. An (2d) lower bound and an O(d2d) upper bound on r(d) are given, r(2) 6 is proved, and the existence of integral Radon partitions, in lattice polytopes having a 1-skeleton with a large stable set of vertices, is established. The computational complexity of deciding if a given set of points in Zd has an integral Radon partition is discussed, and it is shown that if d is fixed, then this problem is in P, while if d is part of the input, it is NP-complete. Key words, abstract convexity, convexity spaces, geometry of numbers, Radon number, Radon partition, lattice polytopes, integer programming, integer lattice AMS(MOS) subject classifications. 52A01, 52A25, 52A35, 52A40, 52A43, 68C25, 90C10