## 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

Review: |

## 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: | 24 Mar 2021 06:48 |

URI: | https://eref.uni-bayreuth.de/id/eprint/63624 |