The slimmest arrangements of hyperplanes: II. Basepointed geometric lattices and Euclidean arrangements

Abstract

We calculate the minimum numbers of k -dimensional flats and cells of any Euclidean d -arrangement of n hyperplanes. The bounds are obtained by calculating lower bounds for the values of the doubly indexed Whitney numbers of a basepointed geometric lattice of rank r with n points. Additional geometric results concern the minimum number of cells of a Euclidean or projective arrangement met by a subspace in general position and the minimum number of non-Radon partitions of a Euclidean point set. We include remarks on the relationship between Euclidean arrangements and basepointed geometric lattices and on the minimum numbers of cells of arrangements with a bounded region.