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.
Previous Page Next Page