Advertisement

On Characterizations of Submanifolds via Smoothness of the Distance Function in Hilbert Spaces

  • David Salas
  • Lionel ThibaultEmail author
Article
  • 21 Downloads

Abstract

The property of continuous differentiability with Lipschitz derivative of the square distance function is known to be a characterization of prox-regular sets. We show in this paper that the property of higher-order continuous differentiability with locally uniformly continuous last derivative of the square distance function near a point of a set characterizes, in Hilbert spaces, that the set is a submanifold with the same differentiability property near the point.

Keywords

Submanifolds Distance function Metric projection Local uniform continuity Diffeomorphism Prox-regular set Hilbert space 

Mathematics Subject Classification

53B25 49J50 41A65 58C20 46C05 

Notes

References

  1. 1.
    Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Gao, D., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications, pp. 99–182. International Press, Somerville (2010)Google Scholar
  2. 2.
    Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352(11), 5231–5249 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the lower-\(C^2\) property. J. Convex Anal. 2(1–2), 117–144 (1995)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ivanov, G.E.: Weak convexity in the senses of Vial and Efimov-Stechkin. Izv. Math. 69, 1113–1135 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cornet, B.: Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96(1), 130–147 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Serea, O.S.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559–575 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cao, T.H., Mordukhovich, B.S.: Optimal control of a nonconvex perturbed sweeping process. J. Differ. Equ. 266(2–3), 1003–1050 (2019)CrossRefzbMATHGoogle Scholar
  8. 8.
    Lewis, A.S., Luke, D.R., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9(4), 485–513 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lewis, A.S., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33(1), 216–234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Poly, J.B.: Fonction distance et sigularités. Bull. Sci. Math. (2me Série) 108(2), 187–195 (1984)zbMATHGoogle Scholar
  11. 11.
    Correa, R., Salas, D., Thibault, L.: Smoothness of the metric projection onto nonconvex bodies in Hilbert spaces. J. Math. Anal. Appl. 457(2), 1307–1322 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Salas, D., Thibault, L.: Characterizations of nonconvex sets with smooth boundary in terms of the metric projection in Hilbert spaces (2018). (pre-print) Google Scholar
  13. 13.
    Holmes, R.B.: Smoothness of certain metric projections on Hilbert space. Trans. Am. Math. Soc. 184, 87–100 (1973)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fitzpatrick, S., Phelps, R.R.: Differentiability of the metric projection in Hilbert space. Trans. Am. Math. Soc. 170(2), 483–501 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Abraham, R., Marsden, J., Ratiu, T.: Manifolds, Tensor Analysis and Applications. Springer, New York (2001)zbMATHGoogle Scholar
  16. 16.
    Field, M.: Differential Calculus and Its Applications. Dover Publications, Mineola (2012)Google Scholar
  17. 17.
    Izzo, A.J.: Locally uniformly continuous functions. Proc. Am. Math. Soc. 122(4), 1095–1100 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nash, J.: Real algebraic manifolds. Ann. Math. Second Ser. 56(3), 405–421 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zarantonello, E.: Projections on convex sets in Hilbert space and spectral theory. Contributions to nonlinear analysis. In: Proceedings of a Symposium Conducted by the Mathematics Research Center, pp. 237–424. The University of Wisconsin-Madison (1971)Google Scholar
  20. 20.
    Canino, A.: On \(p\)-convex sets and geodesics. J. Differ. Equ. 75(1), 118–157 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shapiro, A.: Existence and differentiability of metric projections in Hilbert spaces. SIAM J. Optim. 4(1), 130–141 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Asplund, E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Salas, D., Thibault, L., Vilches, E.: On the smoothness of solutions to projected differential equations. Discrete Contin. Dyn. Syst. Ser. A 39(4), 2255–2283 (2019)CrossRefGoogle Scholar
  24. 24.
    Ioffe, A.D.: An invitation to tame optimization. SIAM J. Optim. 19(4), 1894–1917 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Brokate, M., Krejčí, P.: Optimal control of ode systems involving a rate independent variational inequality. Discrete Contin. Dyn. Syst. B 18, 331–348 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Arroud, C., Colombo, G.: A maximum principle for the controlled sweeping process. Set-Valued Var. Anal. 26(3), 607–629 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Ciencias de la IngenieríaUniversidad de O’HigginsRancaguaChile
  2. 2.CNRS, Institut Montpelliérain Alexander GrothendieckUniversité de MontpellierMontpellierFrance

Personalised recommendations