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Codimension two and three Kneser transversals. (English) Zbl 1392.52003

Summary: Let \(k,d,\lambda \geqslant 1\) be integers with \(d\geqslant \lambda \) and let \(X\) be a finite set of points in \(\mathbb{R}^{d}\). A \((d-\lambda)\)-plane \(L\) transversal to the convex hulls of all \(k\)-sets of \(X\) is called a Kneser transversal. If in addition \(L\) contains \((d-\lambda)+1\) 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 \(\lambda =2,3\). 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 \(d+2(k-\lambda)\) points in \(\mathbb{R}^d\) with \(k-\lambda\geqslant 2\) and \(\lambda =2,3\). We then present a description of Kneser transversals \(L\) of collections of \(d+2(k-\lambda)\) points in \(\mathbb{R}^d\) with \(k-\lambda\geqslant 2\) for \(\lambda =2,3\). We show that either \(L\) is a complete Kneser transversal or it contains \(d-2(\lambda-1)\) 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 \(\lambda =2\) and \(3\)) for \(m(k,d,\lambda)\) defined as the maximum positive integer \(n\) such that every set of \(n\) points (not necessarily in general position) in \(\mathbb{R}^{d}\) admit a Kneser transversal. Finally, by using oriented matroid machinery, we present some computational results (closely related to the stability and unstability notions). We determine the existence of (complete) Kneser transversals for each of the \(246\) different order types of configurations of \(7\) points in \(\mathbb{R}^3\).

MSC:

52A35 Helly-type theorems and geometric transversal theory
52C40 Oriented matroids in discrete geometry
52B55 Computational aspects related to convexity
68-04 Software, source code, etc. for problems pertaining to computer science
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References:

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