Differences of Convex Compact Sets in the Space of Directed Sets, Part I: The Space of Directed Sets

R. Baier, E. Farkhi: Differences of Convex Compact Sets in the Space of Directed Sets, Part I: The Space of Directed Sets
Set-Valued Analysis 9 (3), 217 - 245, 2001

Smart-Link: http://www.ams.org/mathscinet-getitem?mr=1863360
DOI: 10.1023/A:1012046027626
MR Nummer: 1863360
Zentralblattnummer: 1097.49507
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
Mathematics Subject Classification Code: 52A20 (26E25 54C60 65G30 49J53)
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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.

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

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