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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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Article copyright: © Copyright 1986 American Mathematical Society

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