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Differences of convex compact sets in the space of directed sets. I: The space of directed sets. (English) Zbl 1097.49507

Summary: 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 \(\mathbb R^n\) 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 \(\mathbb R^n\) 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 [Set-Valued Anal. 9, 247–272 (2001; Zbl 1097.49508)].

MSC:

49J53 Set-valued and variational analysis
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
54C60 Set-valued maps in general topology
65G30 Interval and finite arithmetic

Citations:

Zbl 1097.49508
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