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Baier, Robert ; Farkhi, Elza M.:
Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets.
In: Set-Valued Analysis.
Bd. 9
(2001)
.
- S. 217-245.
ISSN 0927-6947
DOI: https://doi.org/10.1023/A:1012046027626
| Rez.: |
Abstract
A normed and partially ordered vector space of so-called `directed sets' is constructed, in which the convex cone of all nonempty convex compact sets in |R^n is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n = 1. The directed sets in |R^n are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a `support' function and directed `supporting faces' of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the `support' function and recursively on the directed `supporting faces'. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.
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| Publikationsform: | Artikel in einer Zeitschrift |
|---|---|
| Begutachteter Beitrag: | Ja |
| Zusätzliche Informationen: | Contents:
1. Introduction 2. Preliminaries 2.1 Basic Notations 2.2 Comparison with Other Differences 3. Directed Intervals 3.1 Overview on Known Interval Operations 3.2 Basic Definitions and Operations of Directed Intervals 3.3 Properties of Directed Intervals 4. Directed Sets 4.1 Basic Definitions and Operations of Directed Sets 4.2 A New Metric in the Cone of Convex Compact Sets 4.3 Properties of Directed Sets |
| Keywords: | Directed sets; Directed intervals; Differences of convex sets and their visualization; Embedding of convex compact sets into a vector space; Convex analysis; Interval analysis |
| Fachklassifikationen: | Mathematics Subject Classification Code: 52A20 (26E25 54C60 65G30 49J53) |
| Institutionen der Universität: | Fakultäten > Fakultät für Mathematik, Physik und Informatik Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik V (Angewandte Mathematik) Fakultäten |
| Titel an der UBT entstanden: | Ja |
| Themengebiete aus DDC: | 500 Naturwissenschaften und Mathematik 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Eingestellt am: | 23 Feb 2021 08:11 |
| Letzte Änderung: | 13 Jun 2024 11:50 |
| URI: | https://eref.uni-bayreuth.de/id/eprint/63314 |