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Affine Functions on Compact Convex Sets (unpublished notes) by A.W. Wickstead

By A.W. Wickstead

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Additional resources for Affine Functions on Compact Convex Sets (unpublished notes)

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C. and g If f‘g there is he C(A) with Identify a with 21eP(J1-). c. and convex. Define similarly g : ) ( - 0,, co] by = g and 1311)(0,)\41_ = az. If h separating f and g is continuous and affine, then hI JI„ is the required h. 8. (Michaels) Let 1/411. be a compact Hausdorff space and E a Frechet space. c. and 1(1,x) is a closed convex non-empty subset of E for every -w-eJL, then there is a continuous selection for . : and 2E is defined by (-1,r) = E if As E p continuous affine selection for satisfy all that is required.

0, so we have 0} unless pm = k 1 , thus proving the result. -444 Real valued continuous affine functions. In this section we deal with real valued continuous affine functions defined on a compact simplex. The results of section 1 provide our tools. We prove a variety of extension theorems, and conclude by proving that A(S) has the approximation property. If G is a a-face of S then a function defined on G will be termed affine if its restriction to every face contained in G is affine. 1. Let S be a compact simplex and G a closed 1-00,0D) is a-face of S.

The Bauer simplexes, S, are precisely those compact simplexes affinely homeomorphic to a set of probability measures on a compact Hausdorff space endowed with the weak* topology. Namely S may be identified with P(aO). An alternative characterisation is by means of extension properties for real valued functions defined on the extreme boundary. 3. If K is a compact convex set, the following are equivalent: K is a Bauer simplex. Every continuous function f : ZEK--HfR can be extended to a function of A(K).

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