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
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