Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points

R. Baier, E. Farkhi: Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points
Set-Valued Analysis 15 (2), 185 - 207, 2007

DOI: 10.1007/s11228-006-0038-0
MR Nummer: 2321950
Zentralblattnummer: 1144.28005
Keywords: generalized Steiner selections; Demyanov distance; Aumann integral; Castaing representation; set-valued maps; arithmetic set operations
Mathematics Subject Classification Code: 54C65 (28B20 54C60 26E25 52A20)
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A family of probability measures on the unit ball in |Rn generates a family of generalized Steiner (GS-)points for every convex compact set in |Rn. Such a "rich" family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.


1. Introduction
2. Preliminaries
3. Representations of Sets by Generalized Steiner Points
4. Generalized Steiner Points and Arithmetic Set Operations
5. Regularity Properties of GS-selections
6. Approximate Set-Valued Integration
7. Conclusions

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