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We give geometric proofs of recent results of G. Beer  : the Young-Fenchel correspondence / -> /* 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  about the isometric character of the polarity for closed convex cones.doi:10.1090/s0002-9939-1991-1068129-x fatcat:bafeis5qhzhmdmhpvlkkqkntaa