Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements

摘要: An arrangement of oriented pseudohyperplanes in affined-space defines on its setX a set system (or range space) (X, ?), ? 2x VC-dimensiond natural way: to every cellc the assign subset havingc their positive side, and let be collection all these subsets. We investigate characterize spaces corresponding tosimple arrangements this way; such are calledpseudogeometric, they have property that cardinality is maximum for given VC-dimension. In general, calledmaximum, we show number rangesR?? whichX - R?? also, determines whether space pseudogeometric. Two other characterizations go via simple duality concept "small" subspaces. The correspondence obtained indirectly new characterization uniforom matroids: ?) naturally corresponds uniform matroid rank |X|--d if only VC-dimension isd,R?? impliesX R??, |?| under conditions.