The Dirichlet problem on compact convex sets. (English) Zbl 1486.46007
The following is a rather faithful summary of the Introduction.
Let \(X\) be a compact convex set in a locally convex Hausdorff space and \(\operatorname{ext} X\) the set of all extreme points of \(X\). The Dirichlet problem asks to extend a function \(f\) defined on \(\operatorname{ext} X\) to an affine function \(h\) on \(X\). If \(h\) exists, it is desirable that it preserves as many properties of \(f\) as possible. The existence of an affine continuous extension was characterized in [E. M. Alfsen, Compact convex sets and boundary integrals. Springer-Verlag, Berlin (1971; Zbl 0209.42601), Theorem II.4.5]. For functions of higher Baire classes, a necessary and sufficient condition was presented in [J. Spurný, Isr. J. Math. 173, 403–419 (2009; Zbl 1190.46009), Theorem 3.3].
In this paper the authors generalize these results to the context of vector-valued Baire functions and eliminate the assumption on envelopes which was essential in the above mentioned papers. The main result asserts that, under some mild topological assumption imposed on the set \(\operatorname{ext} X\), a bounded Baire function \(f\) defined on \(\operatorname{ext} X\) is extendable to a so-called strongly affine Baire mapping on \(X\) if and only if \(f\) is annihilated by any boundary measure perpendicular to the space of affine continuous real functions on \(X\). It is easy to see that the condition is necessary and thus the most important part of the proof is concerned with the sufficiency. Four examples witness the sharpness of the results.
Let \(X\) be a compact convex set in a locally convex Hausdorff space and \(\operatorname{ext} X\) the set of all extreme points of \(X\). The Dirichlet problem asks to extend a function \(f\) defined on \(\operatorname{ext} X\) to an affine function \(h\) on \(X\). If \(h\) exists, it is desirable that it preserves as many properties of \(f\) as possible. The existence of an affine continuous extension was characterized in [E. M. Alfsen, Compact convex sets and boundary integrals. Springer-Verlag, Berlin (1971; Zbl 0209.42601), Theorem II.4.5]. For functions of higher Baire classes, a necessary and sufficient condition was presented in [J. Spurný, Isr. J. Math. 173, 403–419 (2009; Zbl 1190.46009), Theorem 3.3].
In this paper the authors generalize these results to the context of vector-valued Baire functions and eliminate the assumption on envelopes which was essential in the above mentioned papers. The main result asserts that, under some mild topological assumption imposed on the set \(\operatorname{ext} X\), a bounded Baire function \(f\) defined on \(\operatorname{ext} X\) is extendable to a so-called strongly affine Baire mapping on \(X\) if and only if \(f\) is annihilated by any boundary measure perpendicular to the space of affine continuous real functions on \(X\). It is easy to see that the condition is necessary and thus the most important part of the proof is concerned with the sufficiency. Four examples witness the sharpness of the results.
Reviewer: Ioan Raşa (Cluj-Napoca)
MSC:
| 46A55 | Convex sets in topological linear spaces; Choquet theory |
| 26A21 | Classification of real functions; Baire classification of sets and functions |
| 54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |
Keywords:
compact convex set; vector-valued Baire function; strongly affine function; Pettis integralReferences:
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