Abstract
Convex functions with continuous epigraph in the sense of Gale and Klée have been studied recently by Auslender and Coutat in a finite-dimensional setting. Here, we provide characterizations of such functionals in terms of the Legendre-Fenchel transformation in general locally convex spaces. Also, we show that the concept of continuous convex sets is of interest in these spaces. We end with a characterization of convex functions on Euclidean spaces with continuous level sets.
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Coutat, P., Volle, M. & Martinez-Legaz, J.E. Convex functions with continuous epigraph or continuous level sets. J Optim Theory Appl 88, 365–379 (1996). https://doi.org/10.1007/BF02192176
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DOI: https://doi.org/10.1007/BF02192176