Continuity of the Fenchel transform of convex functions

Back, Kerry

doi:10.1090/S0002-9939-1986-0845984-5

Continuity of the Fenchel transform of convex functions
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by Kerry Back PDF

Proc. Amer. Math. Soc. 97 (1986), 661-667 Request permission

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

J.-L. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. (9) 52 (1973), 421–441 (1974) (French). MR 500129
John L. Kelley and Isaac Namioka, Linear topological spaces, Graduate Texts in Mathematics, No. 36, Springer-Verlag, New York-Heidelberg, 1976. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, and Kennan T. Smith; Second corrected printing. MR 0394084
Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. MR 283586, DOI 10.1016/0022-247X(71)90200-9
R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966), 81–89. MR 187062
R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966), 46–63. MR 192318, DOI 10.1090/S0002-9947-1966-0192318-X
R. Tyrrell Rockafellar, Conjugate duality and optimization, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. MR 0373611
David W. Walkup and Roger J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229–235. MR 209806, DOI 10.1090/S0002-9939-1967-0209806-6
R. J.-B. Wets, Convergence of convex functions, variational inequalities and convex optimization problems, Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978) Wiley, Chichester, 1980, pp. 375–403. MR 578760
R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32–45. MR 196599, DOI 10.1090/S0002-9947-1966-0196599-8