Items related to Compact Convex Sets and Boundary Integrals
Synopsis
I Representations of Points by Boundary Measures.- § 1. Distinguished Classes of Functions on a Compact Convex Set.- Classes of continuous and semicontinuous, affine and convex functions.- Uniform and pointwise approximation theorems.-Envelopes.-*Grothendieck's completeness theorem.-Theorems of Banach-Dieudonné and Krein-Šmulyan*.- § 2. Weak Integrals, Moments and Barycenters.- Preliminaries and notations from integration theory.-An existence theorem for weak integrals.-Vague density of point-measures with prescribed barycenter.-*Choquet's barycenter formula for affine Baire functions of first class, and a counterexample for affine functions of higher class*.- § 3. Comparison of Measures on a Compact Convex Set.- Ordering of measures.-The concept of dilation for simple measures.-The fundamental lemma on the existence of majorants.-Characterization of envelopes by integrals.-*Dilation of general measures.-Cartier's Theorem*.- § 4. Choquet's Theorem.- A characterization of extreme points by means of envelopes.-The concept of a boundary set.-Hervé's theorem on the existence of a strictly convex function on a metrizable compact convex set.-The concept of a boundary measure, and Mokobodzki's characterization of boundary measures.-The integral representation theorem of Choquet and Bishop - de Leeuw.-A maximum principle for superior limits of 1.s.c. convex functions.-Bishop - de Leeuw's integral theorem relatively to a ?-field on the extreme boundary.-*A counterexample based on the "porcupine topology"*.- § 5. Abstract Boundaries Defined by Cones of Functions.- The concept of a Choquet boundary.-Bauer's maximum principle.-The Choquet-Edwards theorem that Choquet boundaries are Baire spaces.-The concept of a Šilov boundary.-Integral representation by means of measures on the Choquet boundary.- § 6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures.- Ordered convex compacts.-Existence of maximal extreme points.-Characterization of the set of maximal extreme points as a Choquet boundary.-Definition and basic properties of simplicial measures.-Existence of simplicial boundary measures, and the Caratheodory Theorem in ?n.-Decomposition of representing boundary measures into simplicial components.- II Structure of Compact Convex Sets.- § 1. Order-unit and Base-norm Spaces.- Basic properties of (Archimedean) order-unit spaces.-A representation theorem of Kadison.-The vector-lattice theorem of Stone-Kakutani-KreinYosida.-Duality of order-unit and base-norm spaces.- § 2. Elementary Embedding Theorems.- Representation of a closed subspace A of C?(X) as an A(K)-space by the canonical embedding of X in A*.-The concept of an "abstract compact convex" and its regular embedding in a locally convex Hausdorff space.-The connection between compact convex sets and locally compact cones.- § 3. Choquet Simplexes.- Riesz' decomposition property and lattice cones.-Choquet's uniqueness theorem.-Choquet-Meyer's characterizations of simplexes by envelopes.-Edward's separation theorem.-Continuous affine extensions of functions defined on compact subsets of the extreme boundary of a simplex.-Affine Borel extensions of functions defined on the extreme boundary of a simplex.-*Examples of "non-metrizable" pathologies in simplexes.*.- 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary.- Bauer's characterizations of simplexes with closed extreme boundary.-The Dirichlet problem of the extreme boundary.-A criterion for the existence of continuous affine extensions of maps defined on extreme boundaries.- § 5. Order Ideals, Faces, and Parts.- Elementary properties of order ideals and faces.-Extension property and characteristic number.-Archimedean and strongly Archimedean ideals and faces.-Exposed and relatively exposed faces.-Specialization to simplexes.-The concept of a "part", and an inequality of Harnack type.-Characterization of the parts of a simplex in terms of representing measures.-*
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- PublisherSpringer
- Publication date2011
- ISBN 10 3642650104
- ISBN 13 9783642650109
- BindingPaperback
- LanguageEnglish
- Number of pages228
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