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

Title data

Baier, Robert ; Farkhi, Elza:
Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points.
In: Set-Valued Analysis. Vol. 15 (March 2007) Issue 2 . - pp. 185-207.
ISSN 0927-6947
DOI: https://doi.org/10.1007/s11228-006-0038-0

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Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Additional notes: 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
Keywords: Generalized Steiner selections; Demyanov distance; Aumann integral; Castaing representation; Set-valued maps; Arithmetic set operations
Subject classification: Mathematics Subject Classification Code: 54C65 (28B20 54C60 26E25 52A20)
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Result of work at the UBT: Yes
DDC Subjects: 500 Science
500 Science > 510 Mathematics
Date Deposited: 03 Mar 2021 11:01
Last Modified: 19 May 2021 06:10
URI: https://eref.uni-bayreuth.de/id/eprint/63624