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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Continuity of the Fenchel transform of convex functions


Author: Kerry Back
Journal: Proc. Amer. Math. Soc. 97 (1986), 661-667
MSC: Primary 46A55; Secondary 52A07, 90C25
MathSciNet review: 845984
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Abstract: Given a separated dual system $ (E,E')$, the Fenchel transform determines a pairing of the convex functions on $ E$ with the convex functions on $ E'$. This operation is shown to have a continuity property. The result implies that the minimum set of a convex function varies in an upper-semicontinuous way with the function's conjugate. Several convergence concepts for convex functions are discussed. It is shown for each of the two most useful that the Fenchel transform is not a homeomorphism.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0845984-5
PII: S 0002-9939(1986)0845984-5
Article copyright: © Copyright 1986 American Mathematical Society