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Session 10. Generalized Convexity
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Completeness in Minkowski-Rådström-Hörmander spaces |
Hubert Przybycień, Adam Mickiewicz University in Poznań, Poland
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Joint work with J. Grzybowski
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A Minkowski--Rådström-Hörmander space $\widetilde{X}$ is a
quotient space over the family $\mathcal{B} (X)$ of all nonempty
bounded closed convex subsets of a Banach space $X.$ We prove in
that a metric $d_{BP}$ (Bartels--Pallaschke metric) is the strongest
of all complete metrics in the cone $\mathcal{B} (X)$ and Hausdorff
metric $d_H$ is the coarsest of them. Our results follow from for
more general case of a quotient space over an abstract convex cone
$S$ with complete metric $d$. We also extend a definition of
Demyanov's difference (related to Clarke's subdifferential) of
finite dimensional convex sets $A\overset{_{D}}{-}B$ to infinite
dimensional Banach space $X$ and we prove in that Demyanov's metric
generated by such extension, is complete.
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References- Grzybowski J., Przybycień H.,
Completeness in Minkowski--Rådström--Hörmander
spaces, Optimization 2013, online:
\url{http://dx.doi.org/10.1080/02331934.2013.793330
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