Vapnik-Chervonenkis Dimension and (Pseudo-)Hyperplane Arrangements

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 ⊆ 2 of VCdimension 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 also X−R ∈ R, 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 uniform 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.