7. The Euler Characteristic 29 (7.3) Definitions. The real vector space spanned by the functions [A], where A Rd is a compact convex set, is called the algebra of compact convex sets and is denoted K(Rd). Thus a function f K(Rd) is a linear combination f = m i=1 αi[Ai], where the [Ai] Rd are compact convex sets and αi R are real numbers. The real vector space spanned by the functions [A], where A Rd is a closed convex set, is called the algebra of closed convex sets and is denoted C(Rd). Thus a typical function f C(Rd) is a linear combination f = m i=1 αi[Ai], where [Ai] Rd are closed convex sets and αi R are real numbers. We use the term “algebra” since the spaces K(Rd) and C(Rd) are closed under multiplication of functions see Problem 1 below. A linear functional ν : K(Rd) −→ R, resp. ν : C(Rd) −→ R, is called a valuation. Thus ν(αf + βg) = αν(f) + βν(g) for any real α and β and any f, g K(Rd), resp. f, g C(Rd). More generally, we call a valuation any linear transformation K(Rd), C(Rd) −→ V , where V is a real vector space. Valuations will emerge as analogues of “integrals” and “integral transforms” in our combinatorial calculus see Sections 8 and IV.1 for some examples. PROBLEMS. 1◦. Prove that the product fg of functions f, g K(Rd) is a function in K(Rd) and that the product fg of functions f, g C(Rd) is a function in C(Rd). 2◦. Do the functions [A], where A Rd is a non-empty compact convex set, form a basis of K(Rd)? Now we prove the main result of this section. (7.4) Theorem. There exists a unique valuation χ : C(Rd) −→ R, called the Euler characteristic, such that χ([A]) = 1 for every non-empty closed convex set A Rd. Proof. To show that χ must be unique, if it exists, is easy: let f = m i=1 αi[Ai]. Then we must have χ(f) = i:Ai=∅ αi.
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