Inclusion-exclusion complexes for pseudodisk collections
For any finite collection of pseudodisks in the plane (topological disks whose pairwise intersections are either disks or empty) the area can be computed by inclusion-exclusion as \[ \mu(\bigcup B)=\sum_{\sigma\in\Delta}(-1)^{|\sigma|-1}\mu(\bigcap\sigma), \] where \(\Delta\) is an abstract \(2\)-dimensional nerve complex that can be nicely embedded into the interior of the collection. The proof of the result and the algorithm for the computation of such a nerve complex depends on a topological sweep, and therefore they do not generalize to higher dimensions. The paper contains a nice example of a pseudodisk collection that is not equivalent to a collection of geometric disks. It remains an open problem whether analogous results are valid in higher dimensions, as they are for arrangements of geometric balls, due to \textit{H. Edelsbrunner} [``The union of balls and its dual shape,'' in: ``The László Fejes Tóth Festschrift'' (I. Bárány, J. Pach, eds.), Discrete Comput. Geom. 13, No. 3-4, 415-440 (1995; Zbl 0826.68053)].
scientific article; zbMATH DE number 1011310