The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity
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- by Jean-Paul Penot
- Proc. Amer. Math. Soc. 113 (1991), 275-285
- DOI: https://doi.org/10.1090/S0002-9939-1991-1068129-X
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Abstract:
We give geometric proofs of recent results of G. Beer [13]: the Young-Fenchel correspondence $f \to {f^*}$ is bicontinuous on the space of closed proper convex functions on a normed vector space $X$ endowed with the epidistance topology and the polarity operation is continuous on the space of closed convex subsets of $X$ with the bounded Hausdorff topology. Our methods are in the spirit of a famous result due to Walkup and Wets [31] about the isometric character of the polarity for closed convex cones. We also prove that the Hausdorff distance associated with the cosmic distance on the space of convex subsets of a normed vector space induces the bounded -Hausdorff topology. This shows a link between Beer’s results and the continuity results of [2].References
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Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 275-285
- MSC: Primary 54B20; Secondary 46A55, 52A05, 54A20
- DOI: https://doi.org/10.1090/S0002-9939-1991-1068129-X
- MathSciNet review: 1068129