Abstract

Let A⊂RnA\subset\R^n be a closed set and let S⊂RnS\subset\R^n be a set containing AA. In this paper we study a new regularity class for AA, called SS-convexity, introduced by C. Nour, H. Saoud and J. Takche [Regularization via sets satisfying the interior sphere condition, J. Convex Analysis 25(1) (2018)], where an inner approximation of a closed set by sets satisfying the interior sphere condition is given. We prove that this new class covers several known regularity properties including the proximal smoothness, the exterior sphere condition and the union of closed balls. As an application of such results, we provide a new sufficient condition for the equivalence between proximal smoothness and the exterior sphere condition studied previously by C. Nour, R. J. Stern and J. Takche [Proximal smoothness and the exterior sphere condition, J. Convex Analysis 16(2) (2009) 501–514].

Author information

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

Chadi Nour

Dept. of Computer Science and Mathematics, Lebanese American University, P. O. Box 36, Byblos, Lebanon

cnour@lau.edu.lb

Jean Takche

Dept. of Computer Science and Mathematics, Lebanese American University, P. O. Box 36, Byblos, Lebanon

jtakchi@lau.edu.lb

Keywords

  • S-convexity
  • proximal smoothness
  • exterior sphere condition
  • union of closed balls property
  • proximal analysis
  • nonsmooth analysis

Mathematics Subject Classification

49J52, 52A20, 93B27

References

  1. A. Canino: On p-convex sets and geodesics, J. Differential Equations 75(1) (1988) 118–157.https://doi.org/10.1016/0022-0396(88)90132-5Google Scholar
  2. P. Cannarsa, H. Frankowska: Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim. Calc. Var. 12 (2006) 350–370.https://doi.org/10.1051/cocv:2006002Google Scholar
  3. P. Cannarsa, C. Sinestrari: Convexity properties of the minimum time function, Calc. Var. 3 (1995) 273–298.https://doi.org/10.1007/BF01189393Google Scholar
  4. P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control, Birkhäuser, Boston (2004).https://doi.org/10.1007/b138356Google Scholar
  5. F. H. Clarke, Yu. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  6. F. H. Clarke, R. J. Stern, P. R. Wolenski: Proximal smoothness and the lower-C 2 property, J. Convex Analysis 2(1-2) (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  7. G. Colombo, A. Marigonda: Differentiability properties for a class of nonconvex functions, Calc. Var. 25 (2005) 1–31.https://doi.org/10.1007/s00526-005-0352-7Google Scholar
  8. G. Colombo, A. Marigonda, P. R. Wolenski: Some new regularity properties for the minimal time function, SIAM J. Control Optim. 44(6) (2006) 2285–2299.https://doi.org/10.1137/050630076Google Scholar
  9. H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1Google Scholar
  10. C. Nour, H. Saoud, J. Takche: Regularization via sets satisfying the interior sphere condition, J. Convex Analysis 25(1) (2018) 1–19.https://doi.org/10.68381/jca25001Google Scholar
  11. C. Nour, R. J. Stern, J. Takche: Proximal smoothness and the exterior sphere condition, J. Convex Analysis 16(2) (2009) 501–514.https://doi.org/10.68381/jca16027Google Scholar
  12. C. Nour, R. J. Stern, J. Takche: The union of uniform closed balls conjecture, Control and Cybernetics 38(4B) (2009) 1525–1534.Google Scholar
  13. C. Nour, R. J. Stern, J. Takche: The θ-exterior sphere condition, φ-convexity and local semiconcavity, Nonlinear Analysis 73(2) (2010) 573–589.https://doi.org/10.1016/j.na.2010.04.001Google Scholar
  14. C. Nour, R. J. Stern, J. Takche: Validity of the union of uniform closed balls conjecture, J. Convex Analysis 18(2) (2011) 589–600.https://doi.org/10.68381/jca18037Google Scholar
  15. J.-P. Penot: Calculus without Derivatives, Graduate Texts in Mathematics 266, Springer, New York (2013).https://doi.org/10.1007/978-1-4614-4538-8Google Scholar
  16. R. A. Poliquin, R. T. Rockafellar: Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996) 1805–1838.https://doi.org/10.1090/S0002-9947-96-01544-9Google Scholar
  17. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  18. R. T. Rockafellar: Clarke’s tangent cones and the boundaries of closed sets in Rn, Nonlinear Anal. TMA 3 (1979) 145–154.https://doi.org/10.1016/0362-546X(79)90044-0Google Scholar
  19. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  20. A. S. Shapiro: Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim. 4 (1994) 231–259.https://doi.org/10.1137/0804006Google Scholar
  21. C. Sinestrari: Semiconcavity of the value function for exit time problems with nonsmooth target, Commun. Pure Appl. Anal. 3(4) (2004) 757–774.https://doi.org/10.3934/cpaa.2004.3.757Google Scholar

Suggested citation

C. Nour, J. Takche. “A New Class of Sets Regularity.” Journal of Convex Analysis 25 (2018), No. 4, 1059–1074. https://doi.org/10.68381/jca25068

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.