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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2002 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.