| Title:
|
Chebyshev centers in hyperplanes of $c_0$ (English) |
| Author:
|
Veselý, Libor |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
721-729 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give a full characterization of the closed one-codimensional subspaces of $c_0$, in which every bounded set has a Chebyshev center. It turns out that one can consider equivalently only finite sets (even only three-point sets) in our case, but not in general. Such hyperplanes are exactly those which are either proximinal or norm-one complemented. (English) |
| Keyword:
|
Chebyshev centers |
| Keyword:
|
proximinal hyperplanes |
| Keyword:
|
space $c_0$ |
| MSC:
|
41A65 |
| MSC:
|
46B20 |
| MSC:
|
46B25 |
| idZBL:
|
Zbl 1012.41029 |
| idMR:
|
MR1940053 |
| . |
| Date available:
|
2009-09-24T10:55:57Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127758 |
| . |
| Reference:
|
[1] D. Amir: Best simultaneous approximation (Chebyshev centers).Parametric Optimization and Approximation (Oberwolfach 1983), Internat. Ser. Numer. Math. 72, B. Brosowski, F. Deutsch (eds.), Birkhauser-Verlag, Basel, 1985, pp. 19–35. Zbl 0563.41021, MR 0882194 |
| Reference:
|
[2] D. Amir and J. Mach: Chebyshev centers in normed spaces.J. Approx. Theory 40 (1984), 364–374. MR 0740649, 10.1016/0021-9045(84)90011-X |
| Reference:
|
[3] D. Amir, J. Mach and K. Saatkamp: Existence of Chebyshev centers, best $n$-nets and best compact approximants.Trans. Amer. Math. Soc. 271 (1982), 513–524. MR 0654848 |
| Reference:
|
[4] J. Blatter and E. W. Cheney: Minimal projections on hyperplanes in sequence spaces.Ann. Mat. Pura. Appl. 101 (1974), 215–227. MR 0358179, 10.1007/BF02417105 |
| Reference:
|
[5] A. L. Garkavi: The best possible net and the best possible cross section of a set in a normed space.Izv. Akad. Nauk. SSSR 26 (1962), 87–106. (Russian) Zbl 0158.13602, MR 0136969 |
| Reference:
|
[6] R. B. Holmes: A Course in Optimization and Best Approximation. Lecture Notes in Math. 257.Springer-Verlag, 1972. MR 0420367 |
| Reference:
|
[7] L. Veselý: Generalized centers of finite sets in Banach spaces.Acta Math. Univ. Comenian. 66 (1997), 83–115. MR 1474552 |
| Reference:
|
[8] L. Veselý: A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers.Arch. Math (to appear). MR 1967268 |
| Reference:
|
[9] V. N. Zamjatin: The Chebyshev center in hyperspaces of continuous functions.Funktsional’nyj Analiz, vol. 12, A. V. Štraus (ed.), Ul’janovsk. Gos. Ped. Inst., Ul’janovsk, 1979, pp. 56–68. (Russian) MR 0558342 |
| . |
