On the Euler characteristic of finite unions of convex sets |
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Authors: | Beifang Chen |
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Institution: | (1) Department of Mathematics, Massachusetts Institute of Technology, 02139 Cambridge, MA, USA |
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Abstract: | The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces,
its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its
combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such
as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial
Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for
polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting
convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator
functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper. |
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