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Choquet’s theory and the Dirichlet problem. (English) Zbl 1049.31004
The aim of the paper is to illustrate the importance of the notion of convex sets in potential theory and functional analysis. The authors start with a motivation from several different fields arriving at the question whether it is possible to find a measure in the theorem on integral representation which is concentrated only in the set of extremal points. G. Choquet solved the problem in the 1950s founding what is called Choquet’s theory. The rest of the paper is devoted to various aspects of the theory in the more general context of function spaces.
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
52A99 General convexity
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