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# Convex Cones by Benno Fuchssteiner

By Benno Fuchssteiner

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Extra resources for Convex Cones

Example text

Xny xoy ko E X C1 I h 5 C2 - X ~ ~ . . m~ E X X with . THEOREM The aim of t h i s c h a p t e r i s t h e g e n e r a l i z a t i o n o f t h e F i n i t e Decomposition Theorem t o a measure t h e o r e t i c s i t u a t i o n . The r e s u l t which we o b t a i n h e r e i s a r a t h e r powerful g e n e r a l i z a t i o n o f t h e c l a s s i c a l Riesz R e p r e s e n t a t i o n Theorem. We would l i k e t o c a l l t h i s g e n e r a l i z a t i o n t h e Riesz-Konig Theorem s i n c e t h e u n d e r l y i n g i d e a appeared i n a paper o f Heinz Konig [139].

Q I p and t h e Sandwich Theorem g ves us a l i n e a r . Restriction o f that linear functional to X gives the desired h . o Corollary: Let f and g be f u n c t i o n s X + 6 -such t h a t f i s concave ~ and g convex. Then t h e r e i s an a f f i n e f u n c t i o n h w i t h f I h I g i f and - ----- if - only - f I g. Remark : If t h e f u n c t i o n f i s n o t d e f i n e d on a l l o f X t h e n we extend i t t o X b y p u t t i n g f ( x ) = - w whenever X i s o u t s i d e t h e domain o f f The theorem t h e n goes o v e r unchanged.

Here, o f course, by ^f we mean t h e e v a l u a t i o n : f(p) = p(f) set R for all o f sub1 i n e a r p€n . ,n. 1. and l e t F be a subcone of USC(X). Assume n ___--t o be a compact subset of X -such t h a t : supn(f) = supX(f) f o r a l l f E F. Then f o r every l i n e a r u on F with ---p(f) supX(f) for a l l I t h e r e i s a Bore1 p r o b a b i l i t y measure ---p(f) 5 Proof: Take = f J f n la dT T f E F , on n with - for all f -- E F. 1. 2 i n connection w i t h boundaries ( t h e n i n t h e c o r o l l a r y will then be c a l l e d a sup-boundary o r even maxboundary s i n c e s u p n ( f ) = max f ( x ) f o r a l l f E USC(n)).