By A.W. Wickstead
Read or Download Affine Functions on Compact Convex Sets (unpublished notes) PDF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Extra resources for Affine Functions on Compact Convex Sets (unpublished notes)
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. If S is a simplex then there are no non-zero boundary affine dependences on S.
In this section we look at the relationship between faces of K, order ideals in A(K) and quotients of A(K) by such ideals. Throughout this section it will be convenient to consider K embedded in A(K)* = E in the usual manner. If a EA(K) we may also regard a as a (weak*) continuous linear functional on E, this we do with no further explanation. E will be ordered by the usual cone, and will normally be considered with the weak* topology as the dual of A(K), but at times also with the dual norm topology.
8 and any (k F) (k E s F). continuous affine selection of H will satisfy the requirements. 7. Let S be a compact simplex, F a closed face of S and E an ordered Banach space. >0 and ilh it. 3 to the case of functions with values in a (non-ordered) Banach space. 8. Let S be a compact simplex, F a closed face of S and E a Banach space. c. and p(k) for all kE F. p(k) for all kE S. c. and affine. c. ) and affine (as p is concave and strictly positive). If we let H(k) = Q(k) n fe EE : II e p(k) then H(k) is closed and H is affine.
Affine Functions on Compact Convex Sets (unpublished notes) by A.W. Wickstead