Mathematische Annalen

, Volume 365, Issue 3–4, pp 923–967 | Cite as

The NSLUC property and Klee envelope

  • A. Jourani
  • L. ThibaultEmail author
  • D. Zagrodny


A notion called norm subdifferential local uniform convexity (NSLUC) is introduced and studied. It is shown that the property holds for all subsets of a Banach space whenever the norm is either locally uniformly convex or k-fully convex. The property is also valid for all subsets of the Banach space if the norm is Kadec-Klee and its dual norm is Gâteaux differentiable off zero. The NSLUC concept allows us to obtain new properties of the Klee envelope, for example a connection between attainment sets of the Klee envelope of a function and its convex hull. It is also proved that the Klee envelope with unit power plus an appropriate distance function is equal to some constant on an open convex subset as large as we need. As a consequence of obtained results, the subdifferential properties of the Klee envelope can be inherited from subdifferential properties of the opposite of the distance function to the complement of the bounded convex open set. Moreover the problem of singleton property of sets with unique farthest point is reduced to the problem of convexity of Chebyshev sets.

Mathematics Subject Classification

Primary 49J52 46B20 Secondary 52A41 41A50 



We thank both referees for their comments which allowed us to improve the presentation of the paper.


  1. 1.
    Asplund, E.: Farthest points in reflexive locally uniformly rotund Banach spaces. Israel J. Math. 4, 213–216 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asplund, E.: Sets with unique farthest points. Israel J. Math. 5, 201–209 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bandyopadhyay, P.: The Mazur Intersection property for familes of closed bounded convex sets in Banach spaces. Colloq. Math. 63, 45–56 (1992)MathSciNetGoogle Scholar
  4. 4.
    Bandyopadhyay, P., Dutta, S.: Farthest points and the farthest distance map. Bull. Aust. Math. Soc. 71, 425–433 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bauschke, H.H., Macklem, S., Wang, X.: Chebyshev Sets, Klee Sets, and Chebyshev Centers with Respect to Bregman Distances: Recent Results and Open Problems, 1–21 in Fixed-Point Algorithms for Inverse Problems in Sciences and Engineering. Springer Optimization and Its Applications. Springer, New York (2011)zbMATHGoogle Scholar
  7. 7.
    Borwein, J., Fitzpatrick, S.: Existence of nearest points in Banach spaces. Can. J. Math. 4, 702–720 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Dongjian, Lin, Bor-Luh: Ball separation properties in Banach spaces. Rocky Mt. J. Math. 28, 835–873 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cibulka, R., Fabian, M.: Attainment and (sub) differentiability of the supremal convolution of a function and square of the norm. J. Math. Anal. Appl. 393, 632–643 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Deutsch, F.: Best Approximation in Inner Product Spaces. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. Longman scientific and technical (1993)Google Scholar
  12. 12.
    Deville, R., Zizler, V.: Farthest points in \(w^*\)-compact sets. Bull. Aust. Math. Soc. 38, 433–439 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Diestel, J.: Geometry of Banach Spaces-Selected Topics. Springer, Berlin (1975)zbMATHGoogle Scholar
  14. 14.
    Dutta, S.: Generalized subdifferential of the distance function. Proc. Am. Math. Soc. 133, 2949–2955 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Edelstein, M.: Farthest points in uniformly convex Banach spaces. Israel J. Math. 4, 171–176 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Edelstein, M., Lewis, J.: On exposed and farthest points in normed linear spaces. J. Aust. Math. Soc. 12, 301–308 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. CMS Books in Mathrmatics. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  18. 18.
    Fan, K., Glicksberg, I.: Fully convex normed linear spaces. Proc. Natl. Acad. Sci. USA 41, 947–953 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fan, K., Glicksberg, I.: Some geometric properties of the sphere in a normed linear space. Duke Math. J. 25, 553–568 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fitzpatrick, S.: Metric projections and the differentiability of distance functions. Bull. Aust. Math. Soc. 22, 291–312 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Goebel, K., Schöneberg, R.: Moons, bridges, birds \(\cdots \) and nonexpansive mappings in Hilbert space. Bull. Aust. Math. Soc. 17, 463–466 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hiriart-Urruty, J.-B.: La conjecture des points les plus éloignés revisitée. Annales des Sciences Mathématiques du Québec 29, 197–214 (2005)MathSciNetGoogle Scholar
  23. 23.
    Hiriart-Urruty, J.-B.: Potpourri of conjectures and open questions in nonlinear analysis and optimization. SIAM Rev. 49, 255–273 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Holmes, R.B.: Geometrical Functional Analysis and Applications. Springer, New York (1975)CrossRefGoogle Scholar
  25. 25.
    Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Funct. Anal. 266, 1185–1237 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Klee, V.: Convexity of Chebyshev sets. Math. Ann. 142, 292–304 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lau, K.S.: Farthest points in weakly compact sets. Israel J. Math. 22, 168–176 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mazur, S.: \(\ddot{{\rm U}}\)ber schwache Konvergenz in den Ra\(\ddot{{\rm u}}\)men \((L^P)\). Stud. Math. 4, 128–133 (1933)Google Scholar
  29. 29.
    Montesinos, V., Zizler, P., Zizler, V.: Some remarks on farthest points. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A, Matematicas 105, 119–131 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330, p. 584. Springer, Berlin (2006)Google Scholar
  31. 31.
    Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Moreau, J.J.: Fonctionnelles Convexes, Collège de France, 1966, 2nd edn. Tor Vergata University, Roma (2003)Google Scholar
  34. 34.
    Motzkin, T.S., Straus, E.G., Valentine, F.A.: The number of farthest points. Pac. J. Math. 3, 221–232 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Polak, T., Sims, B.: A Banach space which is fully \(2-\)rotund but not locally uniformly rotund. Can. Math. Bull. 26, 118–120 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Schaefer, H.H.: Topological Vector Spaces. Springe, New York (1971). Third Printing CorrectedCrossRefzbMATHGoogle Scholar
  37. 37.
    Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, X.: On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368, 293–310 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Whestphal, U., Schwartz, T.: Farthest points and monotone operators. Bull. Autral. Math. Soc. 58, 75–92 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Wu, Z., Ye, J.J.: Equivalence among various derivatives and subdifferentials of the distance function. J. Math. Anal. Appl. 282, 629–647 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zagrodny, D.: The cancellation law for inf-convolution of convex functions. Stud. Math. 110, 271–282 (1994)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Zagrodny, D.: On the best approximation in real Hilbert spaces. Set-Valued Var. Anal. (submitted)Google Scholar
  43. 43.
    Zizler, V.: On some extremal problems in Banach spaces. Math. Scand. 32, 214–224 (1973)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut de Mathématiques de Bourgogne, UMR 5584 CNRSUniversité de BourgogneDijon CedexFrance
  2. 2.Institut de Mathématiques, UMR 5149 CNRSUniversité Montpellier IIMontpellier CedexFrance
  3. 3.Centro de Modelamiento MatematicoUniversidad de ChileSantiagoChile
  4. 4.Faculty of Mathematics and Natural Sciences, College of Science, Institute of MathematicsCardinal Stefan Wyszynski UniversityWarsawPoland

Personalised recommendations