Directed Sets and Differences of Convex Compact Sets

R. Baier, E. Farkhi: Directed Sets and Differences of Convex Compact Sets
in: Systems modelling and optimization, Proceedings of the 18th IFIP TC7 Conference held in Detroit, Michigan, July 22-25, 1997, P. M. Polis, A. L. Dontchev, P. Kall, I. Lasiecka, A. W. Olbrot (eds.)
Research Notes in Mathematics, Chapman and Hall/CRC, Boca Raton-London-New York-Washington, D.C., 1999, 135 - 143

MR Nummer: 1678692
Zentralblattnummer: 1018.52003
Keywords: convex sets in n dimensions; set-valued maps; set-valued interpolation; interval arithmetic; set-valued analysis
Mathematics Subject Classification Code: 52A20 (49M25 65D05 65G30 93B03 49J53 54C60)
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Abstract:

This paper is devoted to directed sets and differences of convex compact sets. The authors use the specific parametrization of convex compact sets via their support functions and consider the supporting faces as lower-dimensional convex sets. Extending this approach they define a directed set as a pair of mappings that associate to each unit direction an (n-1)-dimensional directed set and a scalar function determining the position of this face in |R^n.
The main differences of the authors' approach to other existing embeddings are that there are no equivalence classes and, secondly, that differences of directed convex sets in |R^n are not real-valued functions of n arguments. The authors provide an application, by giving an example of set-valued interpolation where nonconvex visualizations of directed sets appear as results.

Contents:

1. Introduction
2. Directed Intervals
3. Directed Sets
4. Applications and Numerical Example

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