We prove that (a) in a reflexive space, for any linearly bounded but unbounded closed convex subset the nonsupport functionals are a dense Gδ subset of the polar set, and (b) any nonsemicoercive proper convex lsc [weak*-lsc] function in a [dual] Banach space has a generic [dense Gδ] set of L∞-perturbations which do not attain their infimum. We also characterize the proper convex functions that have inf-nonattaining L∞-perturbations. This results also in a criterion for reflexivity.
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