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
Previous Page Next Page