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

R. Baier, E. Farkhi:
**Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points**

*Set-Valued Analysis*
**15**
(2),
185 - 207,
2007

DOI:
10.1007/s11228-006-0038-0

MR Nummer:
2321950

Zentralblattnummer:
1144.28005

Keywords: generalized Steiner selections; Demyanov distance;
Aumann integral; Castaing representation; set-valued maps; arithmetic set operations

Mathematics Subject Classification Code: 54C65 (28B20 54C60 26E25 52A20)

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## 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.

## 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 |