Abstract
Let B be a finite pseudodisk collection in the plane. By the principle of inclusion-exclusion, the area or any other measure of the union is
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We show the existence of a two-dimensional abstract simplicial complex, χ ⊆ 2B, so the above relation holds even if χ is substituted for 2B. In addition, χ can be embedded in ℝ2 so its underlying space is homotopy equivalent to int ⋃ B, and the frontier of χ is isomorphic to the nerve of the set of boundary contributions.
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This work was supported by the National Science Foundation, under Grant ASC-9200301 and the Alan T. Waterman Award CCR-9118874. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation.
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Edelsbrunner, H., Ramos, E.A. Inclusion-exclusion complexes for pseudodisk collections. Discrete Comput Geom 17, 287–306 (1997). https://doi.org/10.1007/PL00009295
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DOI: https://doi.org/10.1007/PL00009295