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.