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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Abstract:

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.

Contents:

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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