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

Robert Baier, Elza M. Farkhi

Research output: Contribution to journalArticlepeer-review

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 ℝn 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 ℝn 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.

Original languageEnglish
Pages (from-to)217-245
Number of pages29
JournalSet-Valued Analysis
Volume9
Issue number3
DOIs
StatePublished - 2001

Keywords

  • Convex analysis
  • Differences of convex sets and their visualization
  • Directed intervals
  • Directed sets
  • Embedding of convex compact sets into a vector space
  • Interval analysis

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