Integral Representation Theory

This ambitious and substantial monograph, written by prominent experts in the field, presents the state of the art of convexity, with an emphasis on the interplay between convex analysis and potential theory; more particularly, between Choquet theory and the Dirichlet problem. The book is unique and self-contained, and it covers a wide range of applications which will appeal to many readers.

Autorentext

Jaroslav Luke, Jan Malý, Ivan Netuka and Jiri Spurný, Charles University, Prague, Czech Republic.

Zusammenfassung

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