By A.W. Wickstead
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Extra resources for Affine Functions on Compact Convex Sets (unpublished notes)
Sample text
We call E an ordered Banach space if it possesses a closed cone, P. In this case we give A(K,E) the cone f a F A(K,E) : a(k)EP for all k the open ball in E of radius Pc( = P a- . 13,,c will denote and centre the origin, + We look first at the ordered case. E is said to satisfy G(ot ) ( ) if x EX and x E B i implies there exists y E 13,4, -x. 1. -`1 . E satisfies G( 01 ) if and only if A(S,E) satisfies G('-). Suppose E satisfies G( °) and to show that if H a EA(S,E). > 2 E by b >„ a, -a and 11 b II ‘S 04.
S, then 4s = E s so ps (aof) = a(f(s)) for all aEA(K) so -36f(s) = f(s). 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).
Thus -C p E P(K) p,(P) = 11 map is continuous and affine, r -1 is closed. As the resultant (F) is a closed face of P(K). By the Krein-Milman theorem, r-1 (F) = co : k E C E P(K) : u(F) = 11 . As and this latter set is closed and convex, it follows that r-1 (F)C A : k E Cu EP(K) p,(F) = 1 . subset G of a convex set C is a a-face if x, yE C, 0
