8. Application: Convex Sets and Linear Transformations 35

2. Construct an example of compact non-convex sets Ai ⊂

Rn

and real numbers

αi such that

∑k

i=1

αi[Ai] = 0 but

∑k

i=1

αi[T (Ai)] = 0 for some linear transforma-

tion T : Rn −→ Rm.

3. Construct an example of non-compact convex sets Ai ⊂ Rn and real numbers

αi such that

∑k

i=1

αi[Ai] = 0 but

∑k

i=1

αi[T (Ai)] = 0 for some linear transforma-

tion T :

Rn

−→

Rm.

(8.3) Some interesting valuations. Intrinsic volumes. Let vold(A) be the

usual volume of a compact convex set A ⊂

Rd.

The function vold satisfies a number

of useful properties:

(8.3.1) The volume is (finitely) additive: If A1,... , Am ⊂ Rd are compact convex

sets and if α1,... , αm are numbers such that α1[A1] + . . . + αm[Am] = 0, then

α1 vold(A1) + . . . + αm vold(Am) = 0.

(8.3.2) The volume is invariant of

Rd,

that is, orthogonal transfor-

mations and translations: vold

(under)isometries

T (A) = vold(A) for any isometry T : Rd −→ Rd.

(8.3.3) The volume of a compact convex set A ⊂

Rd

with a non-empty interior is

positive.

(8.3.4) The volume in

Rd

is homogeneous of degree d: vold(αA) =

αd

vold(A) for

α ≥ 0.

It turns out that for every k = 0, . . . , d there exists a measure wk on compact

convex sets in Rd, which satisfies properties (8.3.1)–(8.3.3) and which is homoge-

neous of degree k: wk(αA) = αkwk(A) for α 0. These measures are called

intrinsic volumes. For k = d we get the usual volume and for k = 0 we get the

Euler characteristic.

To construct the intrinsic volumes, we observe that the volume can be extended

to a valuation ωd :

K(Rd)

−→ R such that ωd([A]) = vold(A) for any compact

convex set A. Indeed, we define

ωd(f) =

Rd

f(x) dx for f ∈

K(Rd),

where dx is the usual Lebesgue measure on

Rd.

Properties of the integral imply

that ωd(α1f1 + α2f2) = α1ωd(f1) + α2ωd(f2), so ωd is a valuation.

Let L ⊂ Rd be a k-dimensional subspace and let PL be the orthogonal pro-

jection PL : Rd −→ L. Using Theorem 8.1, let us construct a linear transforma-

tion PL : K(Rd) −→ K(L) and hence a valuation ωk,L : K(L) −→ R by letting

ωk,L(f) = ωk

(

PL(f)

)

. Thus, for a compact convex set A ⊂

Rd,

the value of

ωk,L([A]) is the volume of the orthogonal projection of A onto L ⊂

Rd.

The functional ωk,L[A] satisfies (8.3.1) and (8.3.3), it is homogeneous of degree

k, but it is not invariant under orthogonal transformations (although it is invariant