TY - JOUR

T1 - Regularity and integration of set-valued maps represented by generalized Steiner points

AU - Baier, Robert

AU - Farkhi, Elza

PY - 2007/6

Y1 - 2007/6

N2 - 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.

AB - 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.

KW - Arithmetic set operations

KW - Aumann integral

KW - Castaing representation

KW - Demyanov distance

KW - Generalized Steiner selections

KW - Set-valued maps

UR - http://www.scopus.com/inward/record.url?scp=34247850699&partnerID=8YFLogxK

U2 - 10.1007/s11228-006-0038-0

DO - 10.1007/s11228-006-0038-0

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AN - SCOPUS:34247850699

SN - 1877-0533

VL - 15

SP - 185

EP - 207

JO - Set-Valued and Variational Analysis

JF - Set-Valued and Variational Analysis

IS - 2

ER -