Abstract
Any abstract convex cone S with a uniformity satisfying the law of cancellation can be embedded in a topological vector space \(\widetilde{S}\) (Urbański, Bull Acad Pol Sci, Sér Sci Math Astron Phys 24:709–715, 1976). We introduce a notion of a cone symmetry and decompose in Theorem 2.12 a quotient vector space \(\widetilde{S}\) into a topological direct sum of its symmetric subspace \(\widetilde{S}_s\) and asymmetric subspace \(\widetilde{S}_a\). In Theorem 2.19 we prove a similar decomposition for a normed space \(\widetilde{S}\). In section 3 we apply decomposition to Minkowski–Rådström–Hörmander (MRH) space with three best known norms and four symmetries. In section 4 we obtain a continuous selection from a MRH space over ℝ2 to the family of pairs of nonempty compact convex subsets of ℝ2.
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Grzybowski, J., Przybycień, H. & Urbański, R. Decomposition of Minkowski–Rådström–Hörmander Space to the Direct Sum of Symmetric and Asymmetric Subspaces. Set-Valued Var. Anal 21, 201–216 (2013). https://doi.org/10.1007/s11228-013-0231-x
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DOI: https://doi.org/10.1007/s11228-013-0231-x
Keywords
- Quasidifferential
- Abstract convex cone
- Cone of nonempty bounded closed convex sets
- Minkowski–Rådström–Hörmander space
- Hausdorff metric
- Demyanov metric
- Bartels–Pallaschke norm