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.
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