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.
Similar content being viewed by others
References
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)
Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352(11), 5231–5249 (2000)
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)
Ivanov, G.E.: Weak convexity in the senses of Vial and Efimov-Stechkin. Izv. Math. 69, 1113–1135 (2005)
Cornet, B.: Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96(1), 130–147 (1983)
Serea, O.S.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559–575 (2003)
Cao, T.H., Mordukhovich, B.S.: Optimal control of a nonconvex perturbed sweeping process. J. Differ. Equ. 266(2–3), 1003–1050 (2019)
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)
Lewis, A.S., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33(1), 216–234 (2008)
Poly, J.B.: Fonction distance et sigularités. Bull. Sci. Math. (2me Série) 108(2), 187–195 (1984)
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)
Salas, D., Thibault, L.: Characterizations of nonconvex sets with smooth boundary in terms of the metric projection in Hilbert spaces (2018). (pre-print)
Holmes, R.B.: Smoothness of certain metric projections on Hilbert space. Trans. Am. Math. Soc. 184, 87–100 (1973)
Fitzpatrick, S., Phelps, R.R.: Differentiability of the metric projection in Hilbert space. Trans. Am. Math. Soc. 170(2), 483–501 (1982)
Abraham, R., Marsden, J., Ratiu, T.: Manifolds, Tensor Analysis and Applications. Springer, New York (2001)
Field, M.: Differential Calculus and Its Applications. Dover Publications, Mineola (2012)
Izzo, A.J.: Locally uniformly continuous functions. Proc. Am. Math. Soc. 122(4), 1095–1100 (1994)
Nash, J.: Real algebraic manifolds. Ann. Math. Second Ser. 56(3), 405–421 (1952)
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)
Canino, A.: On \(p\)-convex sets and geodesics. J. Differ. Equ. 75(1), 118–157 (1988)
Shapiro, A.: Existence and differentiability of metric projections in Hilbert spaces. SIAM J. Optim. 4(1), 130–141 (1994)
Asplund, E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)
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)
Ioffe, A.D.: An invitation to tame optimization. SIAM J. Optim. 19(4), 1894–1917 (2008)
Brokate, M., Krejčí, P.: Optimal control of ode systems involving a rate independent variational inequality. Discrete Contin. Dyn. Syst. B 18, 331–348 (2013)
Arroud, C., Colombo, G.: A maximum principle for the controlled sweeping process. Set-Valued Var. Anal. 26(3), 607–629 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Salas, D., Thibault, L. On Characterizations of Submanifolds via Smoothness of the Distance Function in Hilbert Spaces. J Optim Theory Appl 182, 189–210 (2019). https://doi.org/10.1007/s10957-019-01473-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-019-01473-3
Keywords
- Submanifolds
- Distance function
- Metric projection
- Local uniform continuity
- Diffeomorphism
- Prox-regular set
- Hilbert space