Abstract

Let A be an open convex subset of a real normed linear space X. We prove that if the boundary of A contains a non-LUR point then there exists a Lipschitz quasiconvex function f from A to R\mathbb{R} not admitting any continuous quasiconvex extension to the whole X.

Author information

Contact details are reproduced from the original publication and may be historical.

Carlo Alberto De Bernardi

Dip. di Matematica per le Scienze Economiche, Finanziarie ed Attuariali, Università Cattolica del Sacro Cuore, Milano, Italy

carloalberto.debernardi@unicatt.it

Libor Veselý

Dip. di Matematica "F. Enriques", Università degli Studi di Milano, Italy

libor.vesely@unimi.it

Keywords

  • Lipschitz function
  • quasiconvex function
  • LUR point
  • exposing functional

Mathematics Subject Classification

46A55, 52A05; 52A99, 26B25

References

  1. K. J. Arrow, A. C. Enthoven: Quasi-concave programming, Econometrica 29 (1961) 779–800.https://doi.org/10.2307/1911819Google Scholar
  2. M. Avriel, W. E. Diewert, S. Schaible, I. Zang: Generalized Concavity, Classics in Applied Mathematics No. 63, reprint of the 1988 original, Society for Industrial and Applied Mathematics, Philadelphia (2010).https://doi.org/10.1007/978-1-4684-7600-2Google Scholar
  3. J. Borwein, V. Montesinos, J. Vanderwerff: Boundedness, differentiability and extensions of convex functions, J. Convex Analysis 13 (2006) 587–602.https://doi.org/10.68381/jca13046Google Scholar
  4. J. Borwein, J. Vanderwerff: On the continuity of biconjugate convex functions, Proc. Amer. Math. Soc. 130 (2002) 1797–1803 (electronic).https://doi.org/10.1090/S0002-9939-01-06249-9Google Scholar
  5. C. A. De Bernardi: A note on the extension of continuous convex functions, J. Convex Analysis 24 (2017) 333–347.https://doi.org/10.68381/jca24023Google Scholar
  6. C. A. De Bernardi: On the extension of continuous quasiconvex functions, J. Optimization Theory Appl. 187 (2020) 421–430.https://doi.org/10.1007/s10957-020-01767-xGoogle Scholar
  7. C. A. De Bernardi, E. Miglierina: A variational approach to the alternating projections method, J. Glob. Optimiation 81 (2021) 323–350.https://doi.org/10.1007/s10898-021-01025-yGoogle Scholar
  8. C. A. De Bernardi, E. Miglierina, E. Molho: Stability of a convex feasibility problem, J. Global Optimization 75/4 (2019) 1061–1077.https://doi.org/10.1007/s10898-019-00806-wGoogle Scholar
  9. C. A. De Bernardi, L. Veselý: Extension of continuous convex functions from subspaces I, J. Convex Analysis 21 (2014) 1065–1084.https://doi.org/10.68381/jca21057Google Scholar
  10. C. A. De Bernardi, L. Veselý: Extension of continuous convex functions from subspaces II, J. Convex Analysis 22 (2015) 101–116.https://doi.org/10.68381/jca22006Google Scholar
  11. C. A. De Bernardi, L. Veselý: Tilings of normed spaces, Canad. J. Math. 69 (2017) 321–337.https://doi.org/10.4153/CJM-2015-057-3Google Scholar
  12. C. A. De Bernardi, L. Veselý: Extendability of continuous quasiconvex functions from subspaces, arXiv: 2212.13789 (2022).https://doi.org/10.1090/proc/16234Google Scholar
  13. C. A. De Bernardi, L. Veselý: On extension of uniformly continuous quasiconvex functions, Proc. Amer. Math. Soc. 151/4 (2023) 1705–1716.https://doi.org/10.1090/proc/16234Google Scholar
  14. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics, Springer, New York (2011).https://doi.org/10.1007/978-1-4419-7515-7Google Scholar
  15. H. Greenberg, W. Pierskalla: A review of quasi-convex functions, Oper. Research 19 (1971) 1553–1570.https://doi.org/10.1287/opre.19.7.1553Google Scholar
  16. V. L. Klee, L. Veselý, C. Zanco: Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces, Studia Math. 120 (1996) 191–204.Google Scholar
  17. J.-P. Penot: What is quasiconvex analysis?, Optimization 47 (2000) 35–110.https://doi.org/10.1080/02331930008844469Google Scholar
  18. L. Veselý, L. Zajíček: On extensions of d.c. functions and convex functions, J. Convex Analysis 17 (2010) 427–440.https://doi.org/10.68381/jca17030Google Scholar

Suggested citation

C. A. De Bernardi, L. Veselý. “Rotundity Properties, and Non-Extendability of Lipschitz Quasiconvex Functions.” Journal of Convex Analysis 30 (2023), No. 1, 329–342. https://doi.org/10.68381/jca30018

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.