Comparative properties of three metrics in the space of compact convex sets. (English) Zbl 0895.90151

Diamond, Phil; Kloeden, Peter; Rubinov, Alexander; Vladimirov, Alexander

Set-Valued Anal. 5, No. 3, 267-289 (1997).

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

Keywords:

Hausdorff metric; difference of convex sets; differentiability of convex-valued maps; convex fuzzy sets