Download Affine Functions on Compact Convex Sets (unpublished notes) by A.W. Wickstead PDF

A(S) such that: A(F)) A(F)) 14c 11 Ta ll II all (a E A(F)). If we construct a continuous affine map tp : such that s—)F /IF is the identity, and define (Ta)(k) = a( 'f k), then it easily verified that T has the desired properties. 17), and let topology induced on A(F)* by be the locally convex ep { i . The natural injection of F into A(F)* is a homeomorphism for the given topology of F and the `r -topology on A(F)*. The topology `1- is metrisable as it is defined by a countable family of semi-norms.

Consider the property: (E) For every compact subset D of there is h with 3I D = h and II 1; 11 = II h . K and every h E C(D) II • Every simplex has this property. If K is metrisable this is equivalent to K being a simplex. 1 with f = , g = II hit . Assume now that K is metrisable and has property (E) , we show that K is a simplex. Let k E K and let p,V be two maximal probability measures representing k. We show that =V , which will prove that K is a simplex. As K is metrisable a t K is a G s and p(aeK) = v (2eK) = 1.

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