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