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 Euter 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.
An Euler IntegralGiven a function X on the class of all compact convex sets of a Euclidean space I~ n with constant value I, Hadwiger [8] uniquely extended Z to the class of finite unions of compact convex sets by finite additivity. Groemer [3], using a similar idea, considered the vector space V(•) of functions generated by indicator functions of compact convex sets of R ", and uniquely extended X to a linear functional on V(X'). The indicator function, or simply indicator, of a subset A of ~" is the function 1A such that la(x)= 1 for x eA and la(x)= 0 for xr Groemer's vector space obviously contains Hadwiger's convex ring of finite unions of compact convex sets. Moreover, it has been shown by Groemer that the Minkowski sum, i.e., vector addition, induces uniquely a commutative multiplication on V(X') which plays an important role in combinatorial geometry,
The geometric, algebraic, and combinatorial explanations of Dobinski's formula are presented by mixed volumes of compact convex sets, Mobius inversion, difference operator, and species. The employed method may be useful in proving some other combinatorial identities.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.