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Transactions of the American Mathematical Society

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Existence of Chebyshev centers, best $ n$-nets and best compact approximants


Authors: Dan Amir, Jaroslav Mach and Klaus Saatkamp
Journal: Trans. Amer. Math. Soc. 271 (1982), 513-524
MSC: Primary 46B20; Secondary 41A46
DOI: https://doi.org/10.1090/S0002-9947-1982-0654848-2
MathSciNet review: 654848
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Abstract: In this paper we investigate the existence and continuity of Chebyshev centers, best $ n$-nets and best compact sets. Some of our positive results were obtained using the concept of quasi-uniform convexity. Furthermore, several examples of nonexistence are given, e.g., a sublattice $ M$ of $ C[0,\,1]$, and a bounded subset $ B \subset M$ is constructed which has no Chebyshev center, no best $ n$-net and not best compact set approximant.


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  • [1] E. M. Alfsen and E. G. Effros, Structure in real Banach spaces, Ann. of Math. (2) 96 (1972), 98-173. MR 0352946 (50:5432)
  • [2] D. Amir, Chebyshev centers and uniform convexity, Pacific J. Math. 77 (1978), 1-6. MR 507615 (80h:46017)
  • [3] D. Amir and Z. Ziegler, Relative Chebyshev centers in normed linear spaces. I, J. Approx. Theory 29 (1980), 235-252. MR 597471 (82c:41030)
  • [4] J. R. Calder, W. P. Coleman and R. L. Harris, Centers of infinite bounded sets in a normed space, Canad. J. Math. 25 (1973), 986-999. MR 0328547 (48:6889)
  • [5] F. Deutsch, J. Mach and K. Saatkamp, Approximation by finite rank operators, J. Approx. Theory (to appear). MR 647847 (84m:47058)
  • [6] H. Fakhoury, Approximation des bornés d'un espace de Banach par des compacts et applications à l'approximation des opérateurs bornés, J. Approx. Theory 26 (1979), 79-100. MR 536717 (80g:47024)
  • [7] M. Feder, On a certain subset of $ {L_1}[0,\,1]$ and non-existence of best approximation in some space of operators, J. Approx. Theory 29 (1980), 170-177. MR 595600 (82a:41018)
  • [8] A. L. Garkavi, The best possible net and the best possible cross section of a set in a normed space, Amer. Math. Soc. Transl. 39 (1964), 111-132.
  • [9] -, The conditional Chebyshev center of a compact set of continuous functions, Math. Notes 14 (1973), 827-831.
  • [10] A. L. Garkavi and V. N. Zamyatin, Conditional Chebyshev centers of a bounded set of continuous functions, Math. Notes 18 (1975), 622-627.
  • [11] R. B. Holmes, $ M$-ideals in approximation theory, Approximation Theory. II, Academic Press, New York, 1976, pp. 391-396. MR 0427927 (55:957)
  • [12] M. I. Kadec and V. N. Zamyatin, Chebyshev centers in the space $ C[a,\,b]$, Teor. Funkciĭ, Funkcional. Anal. i Priložen. 7 (1968), 20-26. (Russian) MR 0268583 (42:3480)
  • [13] K. S. Lau, Approximation by continuous vector-valued functions, Studia Math. 68 (1980), 291-298. MR 599151 (82c:54016)
  • [14] -, On a sufficient condition for proximity, Trans. Amer. Math. Soc. 251 (1979), 343-356. MR 531983 (81b:46033)
  • [15] A. J. Lazar and J. Lindenstrauss, Banach spaces whose duals are $ {L_1}$ spaces and their representing matrices, Acta Math. 126 (1971), 165-193. MR 0291771 (45:862)
  • [16] A. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1-62. MR 0430747 (55:3752)
  • [17] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. No. 65 (1964). MR 0179580 (31:3828)
  • [18] J. Lindenstrauss and D. E. Wulbert, On the classification of the Banach spaces whose duals are $ {L_1}$ spaces, J. Funct. Anal. 4 (1969), 332-349. MR 0250033 (40:3274)
  • [19] J. Mach, Best simultaneous approximation of bounded functions with values in certain Banach spaces, Math. Ann. 240 (1979), 157-164. MR 524663 (80i:41016)
  • [21] -, On the existence of best simultaneous approximation, J. Approx. Theory 25 (1979), 258-265. MR 531415 (80j:41059)
  • [22] -, On the continuity of Chebyshev centers, J. Approx. Theory 29 (1980), 223-230. MR 597469 (83d:46019)
  • [23] J. Mach and J. D. Ward, Approximation by compact operators on certain Banach spaces, J. Approx. Theory 23 (1978), 274-286. MR 505751 (80j:47054)
  • [24] E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238. MR 1529282
  • [25] E. Rozema and P. Smith, Global approximation with bounded coefficients, J. Approx. Theory 16 (1976), 162-174. MR 0404950 (53:8748)
  • [26] K. Saatkamp, Schnitteigenschaften und beste Approximation, Doctoral Dissertation, Bonn, 1979.
  • [27] Z. Semadeni, Banach spaces of continuous functions, Vol. 1, PWN, Warsaw, 1971.
  • [28] P. W. Smith and J. D. Ward, Restricted centers in $ C(\Omega )$, Proc. Amer. Math. Soc. 48 (1975), 165-172. MR 0380227 (52:1127)
  • [29] -, Restricted centers in subalgebras of $ C(X)$, J. Approx. Theory 15 (1975), 54-59. MR 0385420 (52:6282)
  • [30] J. D. Ward, Chebyshev centers in spaces of continuous functions, Pacific J. Math. 52 (1974), 283-287. MR 0350287 (50:2780)
  • [31] V. N. Zamyatin, Relative Chebyshev centers in the space of continuous functions, Soviet Math. Dokl. 14 (1973), 610-614.

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DOI: https://doi.org/10.1090/S0002-9947-1982-0654848-2
Article copyright: © Copyright 1982 American Mathematical Society

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