By A.W. Wickstead
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Additional info for Affine Functions on Compact Convex Sets (unpublished notes)
The "only if" is obvious. “eA(F)1 and B1; = (lc clear that ? I G 1F (k) = f(k)1 . It is F F . fl F. c. and affine and coincides with f on 10. A f F Thus al e10 ;0. 3 al p); 11p. Thus A f I F , and hence B f = Bf et F for all f that are continuous and convex. Suppose F is not a face of S, so that kE F and k = (1/2)k 1 + (1/2)k 2 , k 1 , k2 ES, k i *F. 8. 8. As S is a simplex, 4k dominates all measures on S representing k. But let a EA(S) with a(k 1 ) = 1, a1 µk (f) = 04(1/2 = (1/2)(E k 1 + F 40 and take f = a y °.
1 (3). It follows that f is continuous. Faces of Bauer simplexes are easy to describe and have some nice properties. 5. If S is a Bauer simplex, then a subset F of S is a closed face of S if and only if there is a closed subset J. In D Cae,S such that F = co(D). In this case we have D = particular a closed face of a Bauer simplex is a Bauer simplex. If F is a closed face of S, then Z,I0 = Frs"Ae S is closed. By the Krein-Milman theorem F = co(aeF). If, on the other hand, D('L tS is closed and 0<:X <1, k 1 , k2 E S with Xk i + (1 - X)k 2 E co(D) suppose p, is not supported by D.
3)=4(4). The map taking k to its unique representing boundary measure has all the desired properties. (4)==>( 1 ). 11 r extends to a positive linear operator R : C(K)* ---+A(K)*, and m to a positive linear M : A(K)*---+C(K)* with R(Ma) = a for all a EA(K)*. If a, be A(K)*. R(MaN(Mb)RMa, RMb so exceeds a and b. On the other hand if b then Mc:Ma, Mb so c = R(Mc)R(Ma\eMb). It follows that A(K)* is a lattice, so K is a simplex. A measure p on K is an affine dependence if p(a) = 0 for all a GA(K). 3.