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I am trying to locate a modern account of the problem of determining the number of pieces into which a certain geometric set is divided by given subsets. An example of such a problem could be to count the chambers of a real hyperplane arrangements which was solved by Zaslavsky. I am trying to find an article which has some history of related problems. What I have right now are accounts by Grunbaum written 70's.

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It might be difficult to provide a substantive answer unless you can clarify what you mean by "certain geometric sets" and the "given subsets" ... – Joseph O'Rourke Oct 1 '10 at 21:26
For example, the real space cut by finitely many subspaces such as (pseudo)hyperplanes, spheres. Dissection of open convex sets, polytopes etc. Even dissections of spheres, projective spaces, tori etc. – RP Infinity Oct 1 '10 at 22:32
up vote 3 down vote accepted

Maybe look over the Table of Contents of the 1992 book Arrangements of Hyperplanes by Peter Orlik, Hiroaki Terao to see if what you seek is there. The paper "On the number of arrangements of pseudolines" by Stefan Felsner uses some very nice analysis to improve the known upper bound at the time (1996), and cites earlier relevant work. This may be more algorithmic than you care about, but an authoritative survey as of 2000 was "Arrangements and their applications" by Pankaj Agarwal and Micha Sharir in the Handbook of Computational Geometry. If you are more interested in the combinatorics of, say, pseudosphere arrangements, then perhaps you should look at oriented matroids, e.g., Jurgen Richter-Gebert, and Günter Ziegler, "Oriented Matroids" in the Handbook of Discrete and Computational Geometry, 2004.

Perhaps none of this helps, but it may lead to a sharpening of your question.

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