Jaroslav Lukeš

Integral representation theory : applications to convexity, Banach spaces and potential 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.

On approximation of affine Baire-one functions

It is known (G. Choquet, G. Mokobodzki) that a Baire-one affine function on a compact convex set satisfies the barycentric formula and can be expressed as a pointwise limit of a sequence of continuous affine functions. Moreover, the space of Baire-one affine functions is uniformly closed. The aim of this paper is to discuss to what extent analogous properties are true in the context of general function spaces.In particular, we investigate the function spaceH(U), consisting of the functions continuous on the closure of a bounded open setU⊂ℝm and harmonic onU, which has been extensively studied in potential theory. We demonstrate that the barycentric formula does not hold for the spaceB1b(H(U)) of bounded functions which are pointwise limits of functions from the spaceH(U) and thatB1b(H(U)) is not uniformly closed. On the other hand, every Baire-oneH(U)-affine function (in particular a solution of the generalized Dirichlet problem for continuous boundary data) is a pointwise limit of a bounded sequence of functions belonging toH(U).It turns out that such a situation always occurs for simplicial spaces whereas it is not the case for general function spaces. The paper provides several characterizations of those Baire-one functions which can be approximated pointwise by bounded sequences of elements of a given function space.

Choquet like sets in function spaces

Abstract In convex analysis when studying function spaces of continuous affine functions, notions of a geometrical character like faces, split and parallel faces, exposed or Archimedean faces were investigated in detail by many authors. In this paper we transfer these notions to a more general setting of Choquet theory of abstract function spaces. We prefer a direct functional analytic approach to the treatment of problems instead of using a transfer of a function space to its state space. Methods invoked are based mainly on a measure theory and basic tools of functional analysis and are different from ones using a geometric visualization.

Measure Convex and Measure Extremal Sets

We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened. Received by the editors october 8, 2004; revised 2005-02-09. Research supported in part by the grants GA CR 201/03/0935, GA CR 201/03/D120, GA CR 201/03/1027 and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education. AMS subject classification: 46A55, 52A07.

Simultaneous Solutions of the Weak Dirichlet Problem

The main aim of this paper is a geometrical approach to simultaneous solutions of the abstract weak Dirichlet problem. We answer partially a question from the paper [2] where a similar problem was discussed from a potential-theoretical point of view for the case of function spaces consisting of harmonic functions.