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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity


Author: Jean-Paul Penot
Journal: Proc. Amer. Math. Soc. 113 (1991), 275-285
MSC: Primary 54B20; Secondary 46A55, 52A05, 54A20
MathSciNet review: 1068129
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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].


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1068129-X
PII: S 0002-9939(1991)1068129-X
Keywords: Bounded Hausdorff topology, bounded hemi-convergence, convex sets, conjugacy, cosmic distance, epi-distance, Legendre-Fenchel transform, polarity
Article copyright: © Copyright 1991 American Mathematical Society