Abstract

Let G⊂RnG \subset \mathbb{R}^n be an open convex set which is either bounded or contains a translation of a convex cone with nonempty interior. It is known that, for every modulus ω\omega, every function on GG which is both semiconvex and semiconcave with modulus ω\omega is (globally) C1,ωC^{1,\omega}-smooth. We show that this result is optimal in the sense that the assumption on GG cannot be relaxed. We also present direct short proofs of the above mentioned result and of some its quantitative versions. Our results have immediate consequences concerning (i) a first-order quantitative converse Taylor theorem and (ii) the problem whether f∈C1,ω(G)f\in C^{1,\omega}(G) whenever ff is continuous and smooth in a corresponding sense on all lines. We hope that these consequences are of an independent interest.

Author information

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Keywords

  • ω-semiconvex functions
  • ω-semiconcave functions
  • C1,ω-smooth functions
  • smoothness on all lines
  • converse Taylor theorem
  • strongly α(⋅)-paraconvex functions

Mathematics Subject Classification

26B25; 26B35

References

  1. M. Bačák, J. M. Borwein, A. Eberhard, B. S. Mordukhovich: Infimal convolutions and Lipschitzian properties of subdifferentials for prox-regular functions in Hilbert spaces, J. Convex Analysis 17 (2010) 737–763.https://doi.org/10.68381/jca17049Google Scholar
  2. M. Bardi, I. Capuzzo-Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston (1997).https://doi.org/10.1007/978-0-8176-4755-1Google Scholar
  3. G. O. Berger, P. A. Absil, R. M. Jungers, Y. Nesterov: On the quality of first-order approximation of functions with Hölder continuous gradient, J. Optim. Theory Appl. 185 (2020) 17–33.https://doi.org/10.1007/s10957-020-01632-xGoogle Scholar
  4. M. Bougeard, J.-P. Penot, A. Pommellet: Towards minimal assumptions for the infimal convolution regularization, J. Approx. Theory 64 (1991) 245–270.https://doi.org/10.1016/0021-9045(91)90062-FGoogle Scholar
  5. P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston (2004).https://doi.org/10.1007/b138356Google Scholar
  6. J. Duda, L. Zajíček: Semiconvex functions: representations as suprema of smooth functions and extensions, J. Convex Analysis 16 (2009) 239–260.https://doi.org/10.68381/jca16013Google Scholar
  7. J. Duda, L. Zajíček: Smallness of singular sets of semiconvex functions in separable Banach spaces, J. Convex Analysis 20 (2013) 573–598.https://doi.org/10.68381/jca20034Google Scholar
  8. P. Hájek, M. Johanis: Smooth Analysis in Banach Spaces, Walter de Gruyter, Berlin (2014).https://doi.org/10.1515/9783110258998Google Scholar
  9. J.-B. Hiriart-Urruty, P. Plazanet: Moreau’s decomposition theorem revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 325–338.https://doi.org/10.1016/s0294-1449(17)30028-8Google Scholar
  10. G. E. Ivanov: Continuity of optimal controls in differential games, and some properties of weakly and strongly convex functions (Russian), Mat. Zametki 66 (1999) 816–839; translation in Math. Notes 66 (1999) 675–693.https://doi.org/10.4213/mzm1227https://doi.org/10.1007/BF02674325Google Scholar
  11. M. Johanis: A quantitative version of the Converse Taylor theorem: C k,ω - smoothness, Colloq. Math. 136 (2014) 57–64.https://doi.org/10.4064/cm136-1-6Google Scholar
  12. M. Johanis, L. Zajíček: Smoothness via directional smoothness and Marchaud’s theorem in Banach spaces, J. Math. Analysis Appl. 423 (2015) 594–607.https://doi.org/10.1016/j.jmaa.2014.09.068Google Scholar
  13. A. Jourani, L. Thibault, D. Zagrodny: C 1,ω(·) -regularity and Lipschitz-like properties of subdifferential, Proc. Lond. Math. Soc. (3) 105 (2012) 189–223.https://doi.org/10.1112/plms/pdr062Google Scholar
  14. V. Kryštof: Semiconvex Functions and Their Differences (Czech), Bachelor thesis, Charles University, Prague (2013).Google Scholar
  15. V. Kryštof: Generalized versions of Ilmanen lemma: Insertion of C 1,ω or Cloc 1,ω functions, Comment. Math. Univ. Carol. 59 (2018) 223–231.https://doi.org/10.14712/1213-7243.2015.245Google Scholar
  16. V. Kryštof: Remarks on ω-semiconvex functions, in preparation.Google Scholar
  17. J. M. Lasry, P. L. Lions: A remark on regularization in Hilbert spaces, Israel J. Math. 55 (1986) 257–266.https://doi.org/10.1007/BF02765025Google Scholar
  18. J.-J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
  19. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series No. 28, Princeton University Press, Princeton (1970).Google Scholar
  20. S. Rolewicz: On γ-paraconvex multifunctions, Math. Japon. 24 (1979/80) 293–300.Google Scholar
  21. S. Rolewicz: On the coincidence of some subdifferentials in the class of α(·)-paraconvex functions, Optimization 50 (2001) 353–360.https://doi.org/10.1080/02331930108844568Google Scholar
  22. S. Rolewicz: An extension of Mazur’s theorem on Gâteaux differentiability to the class of strongly α(·)-paraconvex functions, Studia Math. 172 (2006) 243–248.https://doi.org/10.4064/sm172-3-3Google Scholar
  23. L. Zajíček: Differentiability of approximately convex, semiconcave and strongly paraconvex functions, J. Convex Analysis 15 (2008) 1–15.https://doi.org/10.68381/jca15001Google Scholar

Suggested citation

V. Kryštof, L. Zajíček. “Functions on a Convex Set which are both ω-Semiconvex and ω-Semiconcave.” Journal of Convex Analysis 29 (2022), No. 3, 837–856. https://doi.org/10.68381/jca29049

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.