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This monograph presents the state of the art of convexity, with an emphasis to integral representation. The exposition is focused on Choquet's theory of function spaces with a link to compact convex sets. An important feature of the book is an interplay between various mathematical subjects, such as functional analysis, measure theory, descriptive set theory, Banach spaces theory and potential theory. A substantial part of the material is of fairly recent origin and many results appear in the book form for the first time. The text is self-contained and covers a wide range of applications. From the contents: Geometry of convex sets Choquet theory of function spaces Affine functions on compact convex sets Perfect classes of functions and representation of affine functions Simplicial function spaces Choquet's theory of function cones Topologies on boundaries Several results on function spaces and compact convex sets Continuous and measurable selectors Construction of function spaces Function spaces in potential theory and Dirichlet problem Applications

Functional analysis. --- Convex domains. --- Banach spaces. --- Potential theory (Mathematics) --- Integral representations. --- Representations, Integral --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Functions of complex variables --- Generalized spaces --- Topology --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Functional calculus --- Functional equations --- Integral equations --- Convex Analysis. --- Dirichlet Problem. --- Functional Analysis. --- Partial Differential Equation. --- Potential Theory. --- Functional analysis --- Convex domains --- Banach spaces --- Integral representations

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Mathematical analysis --- 51 --- Mathematics --- Functions of real variables. --- Potential theory (Mathematics) --- Topology. --- Potential theory (Mathematics). --- 51 Mathematics