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

Robert Baier, Elza Farkhi

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)185-207
Number of pages23
JournalSet-Valued Analysis
Volume15
Issue number2
DOIs
StatePublished - Jun 2007

Keywords

  • Arithmetic set operations
  • Aumann integral
  • Castaing representation
  • Demyanov distance
  • Generalized Steiner selections
  • Set-valued maps

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