Codimension two and three kneser transversals^∗

Let k, d, λ 1 be integers with d λ and let X be a finite set of points in R^d. A (d − λ)-plane L transversal to the convex hulls of all k-sets of X is called a Kneser transversal. If in addition L contains (d − λ) + 1 points of X, then L is called a complete Kneser transversal. In this paper, we present various results on the existence of (complete) Kneser transversals for λ = 2, 3. In order to do this, we introduce the notions of stability and instability for (complete) Kneser transversals. We first give a stability result for collections of d+ 2(k− λ) points in R^d with k− λ 2 and λ = 2, 3. We then present a description of Kneser transversals L of collections of d + 2(k − λ) points in R^d with k−λ 2 for λ = 2, 3. We show that either L is a complete Kneser transversal or it contains d−2(λ−1) points and the remaining 2(k−1) points of X are matched in k−1 pairs in such a way that L intersects the corresponding closed segments determined by them. The latter leads to new upper and lower bounds (in the case when λ = 2 and 3) for m(k, d, λ) defined as the maximum positive integer n such that every set of n points (not necessarily in general position) in R^d admit a Kneser transversal. Finally, by using oriented matroid machinery, we present some computational results (closely related to the stability and unstability notions).