Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements

Abstract

An arrangement of oriented pseudohyperplanes in affine d-space defines on its set X of pseudohyperplanes a set system (or range space) (X, ℛ), ℛ {square image of or equal to} 2x of VC-dimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let ℛ be the collection of all these subsets. We investigate and characterize the range spaces corresponding to simple arrangements of pseudohyperplanes in this way; such range spaces are called pseudogeometric, and they have the property that the cardinality of ℛ is maximum for the given VC-dimension. In general, such range spaces are called maximum, and we show that the number of ranges R∈ℛ for which X - 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 is d, R∈ℛ implies X - R∈ℛ, and |ℛ| is maximum under these conditions. © 1994 Springer-Verlag New York Inc.