Abstract

We present sufficient conditions ensuring that the Legendre-Young-Fenchel biconjugacy is locally Lipschizian. An application to the continuity of some operations on the space of subsets of an Euclidean space is given.

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Keywords

  • Bipolar
  • conjugacy
  • biconjugate
  • convex set
  • excess
  • Fenchel transform
  • Hausdorff distance
  • Minkowski difference
  • support function

Mathematics Subject Classification

26E25, 49J53; 52A20, 54C60, 52A05

References

  1. D. Azé, J.-P. Penot: Operations on convergent families of sets and functions, Optimization 21 (1990) 521–534.https://doi.org/10.1080/02331939008843576Google Scholar
  2. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Maths 580, Springer-Verlag, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  3. S. Gautier: Affine and eclipsing multifunctions, Numer. Anal. Optim. 11 (1990) 679–699.https://doi.org/10.1080/01630569008816397Google Scholar
  4. E. Giner: Ensembles et fonctions étoilés; applications à l’optimisation et au calcul différentiel généralisé, manuscript, Univ. Toulouse I (1981).Google Scholar
  5. V. L. Levin: Semiconic sets, semihomogeneous functions, and a new duality scheme in convex analysis, Dokl. Math. 55 (1997) 421–423; translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 354 (1997) 597–599.Google Scholar
  6. V. L. Levin: Dual representations of convex sets and Gâteaux differentiability spaces, Set-Valued Anal. 7 (1999) 133–157.https://doi.org/10.1023/A:1008759123278Google Scholar
  7. V. L. Levin: Semiconic duality in convex analysis, Trans. Mosc. Math. Soc. 61 (2000) 197–238.Google Scholar
  8. J.-P. Penot: Variations on the theme of nonsmooth analysis: another subdifferential, in: Nondifferentiable Optimization: Motivations and Applications, Proceedings, Sopron, Hungary, 1984, V. F. Demyanov, D. Pallaschke (eds.), Lecture Notes in Economics and Math. Systems 255, Springer Verlag, Berlin (1985) 41–54.https://doi.org/10.1007/978-3-662-12603-5_5Google Scholar
  9. J.-P. Penot: The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity, Proc. Amer. Math. Soc. 113 (1991) 275–285.https://doi.org/10.1090/S0002-9939-1991-1068129-XGoogle Scholar
  10. J.-P. Penot: On the minimization of difference functions, J. Global Optim. 12 (1998) 373–382.https://doi.org/10.1023/A:1008233531797Google Scholar
  11. J.-P. Penot, P. Terpolilli: Cônes tangents et singularités, C. R. Acad. Sci. Paris 296 (1983) 721–724.Google Scholar
  12. K. Pichard: Variants of the Minkowski difference: continuity properties, preprint, Univ. of Pau, June 2000.Google Scholar
  13. A. P. Ponomarev, N. H. Rozov: Stability and convergence of alternating Pontryagin sums, Vestnik Moscow Univ. Ser. XV Vychisl. Mat. Kibernet. (1978) 82–90 (in Russian).Google Scholar
  14. B. N. Psenicnyj: Leçons sur les jeux différentiels, Cahiers de l’INRIA 4 (1971) 145–226.Google Scholar
  15. R. T. Rockafellar, R. J-B. Wets: Variational Analysis, Springer-Verlag, Berlin (1997).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  16. Y. Sonntag, C. Zălinescu: Scalar convergence of convex sets, J. Math. Anal. Appl. 164 (1992) 219–241.https://doi.org/10.1016/0022-247X(92)90154-6Google Scholar
  17. D. B. Silin: On set-valued differentiation and integration, Set-Valued Anal. 5 (1997) 107–146.https://doi.org/10.1023/A:1008678921420Google Scholar

Suggested citation

J.-P. Penot. “Lipschitzian Behavior of the Fenchel Biconjugacy.” Journal of Convex Analysis 9 (2002), No. 2, 601–607. https://doi.org/10.68381/jca09035

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press.