Some geometric properties of typical compact convex sets in Hilbert spaces

F. de Blasi

Studia Mathematica (1999)

  • Volume: 135, Issue: 2, page 143-162
  • ISSN: 0039-3223

Abstract

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An investigation is carried out of the compact convex sets X in an infinite-dimensional separable Hilbert space , for which the metric antiprojection q X ( e ) from e to X has fixed cardinality n+1 ( n ⊆ ℕ arbitrary) for every e in a dense subset of . A similar study is performed in the case of the metric projection p X ( e ) from e to X where X is a compact subset of .

How to cite

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de Blasi, F.. "Some geometric properties of typical compact convex sets in Hilbert spaces." Studia Mathematica 135.2 (1999): 143-162. <http://eudml.org/doc/216647>.

@article{deBlasi1999,
abstract = {An investigation is carried out of the compact convex sets X in an infinite-dimensional separable Hilbert space , for which the metric antiprojection $q_X(e)$ from e to X has fixed cardinality n+1 ($n ⊆ ℕ$ arbitrary) for every e in a dense subset of . A similar study is performed in the case of the metric projection $p_X(e)$ from e to X where X is a compact subset of .},
author = {de Blasi, F.},
journal = {Studia Mathematica},
keywords = {Hilbert space; typical compact convex sets; typical compact sets; metric antiprojection; metric projection},
language = {eng},
number = {2},
pages = {143-162},
title = {Some geometric properties of typical compact convex sets in Hilbert spaces},
url = {http://eudml.org/doc/216647},
volume = {135},
year = {1999},
}

TY - JOUR
AU - de Blasi, F.
TI - Some geometric properties of typical compact convex sets in Hilbert spaces
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 2
SP - 143
EP - 162
AB - An investigation is carried out of the compact convex sets X in an infinite-dimensional separable Hilbert space , for which the metric antiprojection $q_X(e)$ from e to X has fixed cardinality n+1 ($n ⊆ ℕ$ arbitrary) for every e in a dense subset of . A similar study is performed in the case of the metric projection $p_X(e)$ from e to X where X is a compact subset of .
LA - eng
KW - Hilbert space; typical compact convex sets; typical compact sets; metric antiprojection; metric projection
UR - http://eudml.org/doc/216647
ER -

References

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  1. [1] E. Asplund, Farthest points in reflexive locally uniformly rotund Banach spaces, Israel J. Math. 4 (1966), 213-216. Zbl0143.34904
  2. [2] K. Bartke und H. Berens, Eine Beschreibung der Nichteindeutigkeitsmenge für die beste Approximation in der Euklidischen Ebene, J. Approx. Theory 47 (1986), 54-74. Zbl0619.41020
  3. [3] H. F. Bohnenblust and S. Karlin, Geometrical properties of the unit sphere of Banach algebras, Ann. of Math. 62 (1955), 217-229. Zbl0067.35002
  4. [4] J. M. Borwein and S. Fitzpatrick, Existence of nearest points in Banach spaces, Canad. J. Math. 41 (1989), 702-720. Zbl0668.46006
  5. [5] L. E. J. Brouwer, Beweis der Invarianz der Dimensionenzahl, Math. Ann. 70 (1911), 161-165. Zbl42.0416.02
  6. [6] F. S. De Blasi, On typical compact convex sets in Hilbert spaces, Serdica 23 (1997), 255-268. Zbl0976.46010
  7. [7] F. S. De Blasi and T. Zamfirescu, Cardinality of the metric projection on compact sets in Hilbert space, Math. Proc. Cambridge Philos. Soc. 126 (1999), 37-44. Zbl0923.46022
  8. [8] R. De Ville and V. E. Zizler, Farthest points in w*-compact sets, Bull. Austral. Math. Soc. 38 (1988), 433-439. Zbl0656.46012
  9. [9] A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes in Math. 1543, Springer, Berlin, 1993. Zbl0797.49001
  10. [10] M. Edelstein, Furthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171-176. Zbl0151.17601
  11. [11] P. M. Gruber, Die meisten konvexen Körper sind glatt, aber nicht zu glatt, Math. Ann. 229 (1977), 259-266. Zbl0342.52009
  12. [12] P. M. Gruber, A typical convex surface contains no closed geodesics, J. Reine Angew. Math. 416 (1991), 195-205. Zbl0718.52003
  13. [13] P. M. Gruber, Baire categories in geometry, in: Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.), North-Holland, Amsterdam, 1993, 1327-1346. Zbl0791.52002
  14. [14] V. Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51-63. Zbl0092.11602
  15. [15] K. S. Lau, Farthest points in weakly compact sets, Israel J. Math. 22 (1975), 168-174. Zbl0325.46022
  16. [16] C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. II 3 (1941), 5-7. 
  17. [17] J. C. Oxtoby, Measure and Category, Grad. Texts in Math. 2, Springer, New York, 1971. 
  18. [18] E. T. Poulsen, Convex sets with dense extreme points, Amer. Math. Monthly 66 (1959), 577-578. Zbl0104.08401
  19. [19] R. Schneider, On the curvature of convex bodies, Math. Ann. 240 (1979), 177-181. Zbl0379.52004
  20. [20] R. Schneider and J. A. Wieacker, Approximation of convex bodies by polytopes, Bull. London Math. Soc. 13 (1981), 149-156. Zbl0421.52005
  21. [21] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Grundlehren Math. Wiss. 171, Springer, New York, 1970. Zbl0197.38601
  22. [22] S. B. Stečkin [S. B. Stechkin], Approximation properties of sets in normed linear spaces, Rev. Math. Pures Appl. 8 (1963), 5-18 (in Russian). 
  23. [23] J. A. Wieacker, The convex hull of a typical compact set, Math. Ann. 282 (1988), 637-644. Zbl0636.52004
  24. [24] T. Zamfirescu, Nearly all convex bodies are smooth and strictly convex, Monatsh. Math. 103 (1987), 57-62. Zbl0607.52002
  25. [25] T. Zamfirescu, The nearest point mapping is single valued nearly everywhere, Arch. Math. (Basel) 54 (1990), 563-566. Zbl0715.54013
  26. [26] T. Zamfirescu, Baire categories in convexity, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 139-164. Zbl0780.52003
  27. [27] N. V. Zhivkov, Compacta with dense ambiguous loci of metric projection and antiprojection, Proc. Amer. Math. Soc. 123 (1995), 3403-3411. Zbl0842.41024
  28. [28] N. V. Zhivkov, Densely two-valued metric projections in uniformly convex Banach spaces, Set-Valued Anal. 3 (1995), 195-209. Zbl0830.41028

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