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Comparative properties of three metrics in the space of compact convex sets. (English) Zbl 0895.90151
Summary: Along with the Hausdorff metric, we consider two other metrics on the space of convex sets, namely, the metric induced by the Demyanov difference of convex sets and the Bartels-Pallaschke metric. We describe the hierarchy of these three metrics and of the corresponding norms in the space of differences of sublinear functions. The completeness of corresponding metric spaces is demonstrated. Conditions of differentiability of convex-valued maps of one variable with respect to these metrics are proved for some special cases. Applications to the theory of convex fuzzy sets are given.

MSC:
90C25 Convex programming
49J52 Nonsmooth analysis
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
47L07 Convex sets and cones of operators
58C06 Set-valued and function-space-valued mappings on manifolds
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