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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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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

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