## Abstract

A family of probability measures on the unit ball in ℝ^{n} generates a family of generalized Steiner (GS-)points for every convex compact set in ℝ^{n}. 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 language | English |
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Pages (from-to) | 185-207 |

Number of pages | 23 |

Journal | Set-Valued Analysis |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2007 |

## Keywords

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