By A.W. Wickstead
Read Online or Download Affine Functions on Compact Convex Sets (unpublished notes) PDF
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Extra resources for Affine Functions on Compact Convex Sets (unpublished notes)
Let F be a face of a compact simplex. F is split if and only if F is norm closed. 5 shows that if F is norm closed then so is F. We identify the simplex, S, with the state space of A(S). Fix k E S, and let k 1 = sup lmEF : “. 1ci which exists as S - S = A(S)* is a complete vector lattice. If m E F, let m = Xmkm , 0‘Xm , km E F. If k then X,m , 1, for k - m = p,n with p O and n E S. Now we may write A mkm + un 1 “ ) '1 m "" where X + p, = 1 by the uniqueness property of the definition m k = m + 421 = ( A m +µ)( of a base.
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. If S is a simplex then there are no non-zero boundary affine dependences on S. We shall denote by gic the unique boundary measure representing k whenever S is a simplex. 4. c. c. concave function. If gv f g then f g,R,and if f < g then f< R. c. and affine, so attains its infimum on -4S tby the Krein-Milman theorem).
Let S be a compact simplex and E a Banach space. Let 1. : S---)2 E be affine and let / (k) be closed for all k E S. c. and satisfies I c. Then there is a continuous affine selection for + Bp. -33- 2 Bauer simplexes. We shall look in this section at one special case that has inspired much of the general theory. IPA is a compact Hausdorff space, then M(a) is a lattice, so the base of the positive cone in it, P(SL), is a simplex. When endowed with the weak* topology as the dual of C(JL), this simplex is compact.