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.

A1 A

A

A

4

3

2

B3

B4

B

1

B2

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.