Abstract
An arrangement of oriented pseudohyperplanes in affined-space defines on its setX of pseudohyperplanes a set system (or range space) (X, ℛ), ℛ ⊑ 2x of VC-dimensiond in a natural way: to every cellc in the arrangement assign the subset of pseudohyperplanes havingc on their positive side, and let ℛ be the collection of all these subsets. We investigate and characterize the range spaces corresponding tosimple arrangements of pseudohyperplanes in this way; such range spaces are calledpseudogeometric, and they have the property that the cardinality of ℛ is maximum for the given VC-dimension. In general, such range spaces are calledmaximum, and we show that the number of rangesR∈ℛ for whichX - R∈ℛ also, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and “small” subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniforom oriented matroids: a range space (X, ℛ) naturally corresponds to a uniform oriented matroid of rank |X|—d if and only if its VC-dimension isd,R∈ℛ impliesX - R∈ℛ, and |ℛ| is maximum under these conditions.
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Part of this work was done while the first author was a member of the Graduiertenkolleg “Algorithmische Diskrete Mathematik,” supported by the Deutsche Forschungsgemeinschaft, Grant We 1265/2-1. Part of this work has been supported by the German-Israeli Foundation for Scientific Research and Development (G.I.F.).
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Gärtner, B., Welzl, E. Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements. Discrete Comput Geom 12, 399–432 (1994). https://doi.org/10.1007/BF02574389
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DOI: https://doi.org/10.1007/BF02574389