8. Application: Convex Sets and Linear Transformations 33

8. Application: Convex Sets and Linear Transformations

As an application of the Euler characteristic, we demonstrate an interesting behav-

ior of collections of compact convex sets under linear transformations.

(8.1) Theorem. Let T : Rn −→ Rm be a linear transformation. Then there exists

a linear transformation T : K(Rn) −→ K(Rm) such that T ([A]) = [T (A)] for any

compact convex set A ⊂ Rn.

Proof. Clearly, if A ⊂

Rn

is a compact convex set, then T (A) ⊂

Rm

is also a

compact convex set. Let us define a function G :

Rn

×

Rm

−→ R, where

G(x, y) =

1 if T (x) = y,

0 if T (x) = y.

Let f ∈ K(Rn) be a function. We claim that for every y ∈ Rm the function

gy(x) = G(x, y)f(x) belongs to the space K(Rn). Indeed, if

(8.1.1) f =

k

i=1

αi[Ai],

where αi ∈ R and Ai ⊂ Rn are compact convex sets, then

(8.1.2) gy =

k

i=1

αi[Ai ∩ T

−1(y)],

where T −1(y) is the aﬃne subspace that is the inverse image of y. Hence χ(gy) is

well defined and we define h = T (f) by the formula h(y) = χ(gy). We claim that

h ∈ K(Rm). Indeed, for f as in (8.1.1), the function gy is given by (8.1.2) and

h(y) =

i∈I

αi, where I = i : Ai ∩ T

−1

(y) = ∅ .

However, Ai ∩ T −1(y) = ∅ if and only if y ∈ T (Ai), so

(8.1.3) h =

i∈I

αi[T (Ai)].

Therefore, h = T (f) ∈

K(Rm)

and the transformation T is well defined. We see

that T is linear since for f = α1f1 + α2f2 we get

gy(x) = α1g1,y(x) + α2g2,y(x),

where

gy(x) = G(y, x)f(x), g1,y = G(y, x)f1(x) and g2,y = G(y, x)f2(x).

Since χ is a linear functional (see Theorem 7.4), h(y) = α1h1(y) + α2h2(y), where

h = T (f), h1 = T (f1) and h2 = T (f2). It follows from (8.1.3) that T [A] = [T (A)].

In particular, linear dependencies among the indicators of compact convex sets

are preserved by linear transformations.