Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1986-0845984-5
MathSciNet review: 845984
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] J.-L. Joly, Une famille de topologies sur l'ensemble des fonctions convexes pour lesquelles la polarite est bicontinue, J. Math. Pures Appl. 52 (1973), 421-441. MR 0500129 (58:17826)
  • [2] J. L. Kelley and I. Namioka, Linear topological spaces, Springer-Verlag, New York, 1973. MR 0394084 (52:14890)
  • [3] U. Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518-535. MR 0283586 (44:817)
  • [4] R. T. Rockafellar, Extension of Fenchel's duality theorem for convex functions, Duke Math. J. 33 (1966), 81-89. MR 0187062 (32:4517)
  • [5] -, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966), 46-53. MR 0192318 (33:544)
  • [6] -, Conjugate duality and optimization, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 16, SIAM, Philadelphia, Pa., 1974. MR 0373611 (51:9811)
  • [7] D. Walkup and R. J.-B. Wets, Continuity of some convex-cone valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229-235. MR 0209806 (35:702)
  • [8] R. J.-B. Wets, Convergence of convex functions, variational inequalities and convex optimization problems, Variational Inequalities and Complementarity Problems (R. Cottle, F. Gianessi and J.-L. Lions, eds.), Wiley, London, 1980, pp. 375-403. MR 578760 (83a:90140)
  • [9] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32-45. MR 0196599 (33:4786)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A55, 52A07, 90C25

Retrieve articles in all journals with MSC: 46A55, 52A07, 90C25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0845984-5
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society