34 I. Convex Sets at Large
(8.2) Corollary. Let T :
be a linear transformation, let A1,... , Ak
be compact convex sets in
and let α1,... , αk be numbers such that
α1[A1] + . . . + αk[Ak] = 0.
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.
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].
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 ∈
such that (α1f1 + α2f2) g = α1(f1 g) +
α2(f2 g) for any f1,f2,g ∈
and any α1,α2 ∈ R and such that [A1] [A2] =
[A1 + A2] for any compact convex sets A1,A2 ⊂ Rd.