Oriented matroids and hyperplane transversals (Q1915393)

The following theorem is proved: A family \({\mathcal A}\) of compact connected sets in \(R^d\) has a hyperplane transversal if and only if for some \(k\), \(0\leq k<d\), there exists an acyclic oriented matroid of rank \(k+1\) such that every \(k+2\) sets have a \(k\)-dimensional transversal meeting consistently with this oriented matroid. Thus the existence of a hyperplane transversal (a subtle geometric property) has an entirely combinatorial characterization (in terms of abstract oriented matroids). The result is a strong generalization of Hadwiger's theorem about line transversals for disjoint convex sets in the plane, extending previous results for separated families of sets (Goodman and Pollack) and for realizable oriented matroids (Pollack and Wenger). The method of proof is interesting: it establishes and uses two Borsuk-Ulam type theorems for mappings resp. non-surjective mappings into the nerve complex of an acyclic oriented matroid. (This nerve complex coincides with the complex of acyclic sets, which by \textit{P. H. Edelman} [J. Comb. Theory, Ser. B 36, 26-31 (1984; Zbl 0542.05023)] has the homotopy type of a sphere).