Vertical decomposition of arrangements of hyperplanes in four dimensions

Abstract

We show that, for any collection ℋ of n hyperplanes in ℜ4, the combinatorial complexity of the vertical decomposition of the arrangement A(ℋ) of ℋ is O(n4 log n). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them. © 1995 Springer-Verlag New York Inc.