34 I. Convex Sets at Large (8.2) Corollary. Let T : Rn −→ Rm be a linear transformation, let A1,... , Ak be compact convex sets in Rn and let α1,... , αk be numbers such that α1[A1] + . . . + αk[Ak] = 0. Then α1[T (A1)] + . . . + αk[T (Ak)] = 0. Proof. We apply the transformation T to both sides of the identity α1[A1]+ . . . + αk[Ak] = 0. Corollary 8.2 is trivial for invertible linear transformations T but becomes much less obvious for projections see Figure 9. A A A A 1 4 3 2 B 3 B 4 B 1 B 2 Figure 9. Four convex sets A1, A2, A3, A4 such that [A4] = [A1] + [A2]−[A3] and their projections B1, B2, B3, B4. We observe that [B4] = [B1] + [B2] [B3]. PROBLEMS. 1. Prove that the Minkowski sum of compact convex sets is a compact convex set and that there exists a commutative and associative operation f g, called a convolution, for functions f, g K(Rd) such that (α1f1 + α2f2) g = α1(f1 g) + α2(f2 g) for any f1,f2,g K(Rd) and any α1,α2 R and such that [A1] [A2] = [A1 + A2] for any compact convex sets A1,A2 Rd.
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