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On Cell Complexities in Hyperplane Arrangements # Boris Aronov + Micha Sharir #
 

Summary: On Cell Complexities in Hyperplane Arrangements #
Boris Aronov + Micha Sharir #
July 14, 2003
Abstract
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, MatouŸsek, 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 [4]:
Theorem 1.1. The complexity of m distinct cells in an arrangement of n hyperplanes in d di­

  

Source: Aronov, Boris - Department of Computer Science and Engineering, Polytechnic Institute of New York University

 

Collections: Computer Technologies and Information Sciences