Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements

Abstract

An arrangement of oriented pseudohyperplanes in affined-space defines on its setX of pseudohyperplanes a set system (or range space) (X, ?), ? ? 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 ? 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 ? is maximum for the given VC-dimension. In general, such range spaces are calledmaximum, and we show that the number of rangesR?? for whichX - 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, ?) naturally corresponds to a uniform oriented matroid of rank |X|--d if and only if its VC-dimension isd,R?? impliesX - R??, and |?| is maximum under these conditions.