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We give geometric proofs of recent results of G. Beer [13] : the Young-Fenchel correspondence f→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]

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