Summary: On Cell Complexities in Hyperplane Arrangements #
Boris Aronov + Micha Sharir #
July 14, 2003
We derive improved bounds on the complexity, i.e., the total number of faces of all dimen
sions, of many cells in arrangements of hyperplanes in higher dimensions, and use these bounds
to obtain a very simple proof of an earlier bound, due to Aronov, Matousek, and Sharir, on the
sum of squares of cell complexities in such an arrangement.
1 Complexity of Many Cells
In this note we consider collections of cells in arrangements of hyperplanes in ddimensional space.
The complexity of a cell is the total number of faces of all dimensions appearing on its boundary. The
complexity of a collection of cells is the sum of the complexities of its members. The main goal is to
derive sharp bounds on the maximum complexity of any collection of m arbitrary distinct cells in an
arrangement of n hyperplanes in R d . For more extensive background on the ``many cells'' problem
considered in this paper, see [8, 12]. A variety of bounds have been derived in [1, 4,5,7,10,11]. The
problem has also been extended to the case of many cells in arrangements of curves or surfaces;
see, e.g., [2, 3, 7, 9].
The main result of the paper improves upon the previous bound O(m 1/2 n d/2 log (#d/2#-1)/2 n)
given in :
Theorem 1.1. The complexity of m distinct cells in an arrangement of n hyperplanes in d di