Abstract
The closure of preimages (inverse images) of metric projection mappings to a given set in a Hilbert space are investigated. In particular, some properties of fibers over singletons (level sets or preimages of singletons) of the metric projection are provided. One of them, a sufficient condition for the convergence of minimizing sequence for a giving point, ensures the convergence of a subsequence of minimizing points, thus the limit of the subsequence belongs to the image of the metric projection. Several examples preserving this sufficient condition are provided. It is also shown that the set of points for which the sufficient condition can be applied is dense in the boundary of the preimage of each set from a large class of subsets of the Hilbert space. As an application of obtained properties of preimages we show that if the complement of a nonconvex set is a countable union of preimages of convex closed sets then there is a point such that the value of the metric projection mapping is not a singleton. It is also shown that the Klee result, stating that only convex closed sets can be weakly closed Chebyshev sets, can be obtained for locally weakly closed sets.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asplund, E.: C̆ebys̆ev sets in Hilbert spaces. Trans. Amer. Math. Soc. 144, 235–240 (1969)
Balaganskii, V.S., Vlasov, L.P.: The problem of convexity of Chebyshev sets. Russ. Math. Surv. 51(6), 1127–1190 (1996)
Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303, 1–14 (2005)
Bosznay, A.P.: A remark on a problem of Goebel. Ann. Univ. Set. Budapest Eotvos Sect. Math. 28, 143–145 (1985)
Borwein, J.M., Fitzpatrick, S.P., Giles, J.R.: The Differentiability of real functions on normed linear space using generalized subgradients. J. Math. Anal. Appl. 128, 512–534 (1987)
Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer (2005)
Borwein, J.M.: Proximality and Chebyshev sets. Optimization letters, 21–32 (2007)
Correa, R., Jofré, A., Thibault, L.: Characterization of Lower Semicontinuous Convex Functions. Proc. Amer. Math. Soc. 116(1), 67–72 (1992)
Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. Longman Scintific and Technical (1993)
Diestel, J.: Geometry of Banach spaces-selected topics, Springer- Verlag Lecture Notes 485 (1975)
Dutta, S.: Generalized Subdifferential of the distance function. Proc. Amer. Math. Soc., 133(10), 2949–2955 (2005)
Fabian, M., Habala, P., Hajek, P., Montesinos, V., Zizler, V.: Banach Space Theory, The Basis for Linear and Nonlinear Analysis. Springer, New York (2011)
Fitzpatrick, S.: Metric projections and the differentiability of distance functions. Bull. Austral. Math. Soc. 22, 291–312 (1980)
Fitzpatrick, S.: Differentiation of real-valued functions and continuity of metric projections. Proc. Amer. Math. Soc. 91(4), 544–548 (1984)
Fitzpatrick, S.: Nearest points to closed sets and directional derivatives of distance functions. Bull. Austral. Math. Soc. 39, 233–238 (1989)
Friedman, A.: Foundations of Modern Analysis. Dover Publications Inc, New York (1982)
Goebel, K., Schöneberg, R.: Moons, bridges, birds... and nonexpasive mappings in Hilbert space. Bull. Austral. Math. Soc. 17, 463–466 (1977)
Giles, J.R.: Differentiability of distance functions and a proximal property inducing convexity. Proc. Amer. Math. Soc. 104, 458–464 (1988)
Giles, J.R.: A distance function property implying differentiability. Bull. Austral. Math. Soc. 39, 59–70 (1989)
Hiriart-Urruty, J.-B.: New concepts in nondifferentiable programming. Mémoires de la S.M.F., tome 60, 57–85 (1979)
Hiriart-Urruty, J.-B.: Ensembles de Tchebychev vs. ensembles convexes: Létat de lart vu via lanalyse convexe non lisse. Ann. Sci. Math. Québec 22, 4762 (1998)
Hiriart-Urruty, J.-B.: Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization. SIAM Review 49, 255–273 (2007)
Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Fun. Anal. 266, 1185–1237 (2014)
Klee, V.: Convexity of Chebyshev sets. Math. Annalen 142, 292–304 (1961)
Klee, V.L.: Remarks on nearest points in normed linear spaces, in Proc. Colloquium on Convexity (Copenhagen 1965), 168176, Kobenhavns Univ. Mat. Inst., Copenhagen (1967)
Klee, V.L.: Reproduced with comments by B. Grünbaum, Unsolved problems in intuitive geometry. http://www.math.washington.edu/ (1960/2010)
Lau, K.-S.: Almost Chebyshev Subspaces. J. Approx. Theory 21, 319–327 (1977)
Lau, K.-S.: On almost Chebyshev subsets in reflexive Banach spaces. Indiana Univ. Math. J. 27, 791–795 (1978)
Poliquin, R. A., Rockafellar, R. T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352, 5231–5249 (2000)
Ricceri, B.: A conjecture implying the existence of non-convex Chebyshev sets in infinite-dimensional Hilbert spaces. Le Matematiche LXV, 193–199 (2010). – Fasc. II
Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)
Westphal, U., Frerking, J.: On a property of metric projections onto closed subsets of Hilbert spaces. Proc. Amer. Math. Soc. 105, 644–651 (1989)
Vlasov, L.P.: Chebyshev sets and approximatively convex sets. Math. Notes 2, 600–605 (1967)
Vlasov, L.P.: Almost convexity and Chebyshev sets. Math. Notes 8, 776–779 (1970)
Vlasov, L.P.: Approximative properties of sets in normed linear spaces. Russian Math. Surveys 28(174), 3–66 (1973)
Wang, X.: On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368, 293–310 (2010)
Yosida, K.: Functional Analysis, 6th Edn. Springer, Berlin (1980)
Zhivkov, N.V.: Metric projections and antiprojections in strictly convex normed spaces. C.R. Acad. Bulgare Sci. 31(4), 369–372 (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Professor Lionel Thibault to honor great man and great mathematician. Thank You Lionel for the journeys in mathematics which we have gone together.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zagrodny, D. On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces. Set-Valued Var. Anal 23, 581–612 (2015). https://doi.org/10.1007/s11228-015-0350-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-015-0350-7
Keywords
- Inverse mapping of metric projection
- Chebyshev sets
- Best approximation
- Convexity
- Differentiability of the distance function
- Concavity of the distance function