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

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

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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.