Geodesic
Antiproximinal sets in spaces of continuous functions
Matematičeskie zametki, Tome 60 (1996) no. 5, pp. 643-657.

Voir la notice de l'article provenant de la source Math-Net.Ru

Closed convex bounded antiproximinal bodies are constructed in the infinite-dimensional spaces $C(Q)$, $C_0(T)$, $L_\infty(S,\Sigma,\mu)$ and $B(S)$, where $Q$ is a topological space and $T$ is a locally compact Hausdorff space. It is shown that there are no closed bounded antiproximinal sets in Banach spaces with the Radon–Nikodym property. 
@article{MZM_1996_60_5_a0,
     author = {V. S. Balaganskii},
     title = {Antiproximinal sets in spaces of continuous functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--657},
     publisher = {mathdoc},
     volume = {60},
     number = {5},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a0/}
}
TY  - JOUR
AU  - V. S. Balaganskii
TI  - Antiproximinal sets in spaces of continuous functions
JO  - Matematičeskie zametki
PY  - 1996
SP  - 643
EP  - 657
VL  - 60
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a0/
LA  - ru
ID  - MZM_1996_60_5_a0
ER  - 
%0 Journal Article
%A V. S. Balaganskii
%T Antiproximinal sets in spaces of continuous functions
%J Matematičeskie zametki
%D 1996
%P 643-657
%V 60
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a0/
%G ru
%F MZM_1996_60_5_a0
V. S. Balaganskii. Antiproximinal sets in spaces of continuous functions. Matematičeskie zametki, Tome 60 (1996) no. 5, pp. 643-657. http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a0/

[1] Edelstein M., Thompson A. C., “Some results on nearest points and support properties of convex sets in $c_0$”, Pacific J. Math., 40:3 (1972), 553–560 | MR | Zbl

[2] Kobzash S., “Vypuklye antiproksiminalnye mnozhestva v prostranstvakh $c_0$ i $c$”, Matem. zametki, 17:3 (1975), 449–457 | MR | Zbl

[3] Cobzaş S., “Antiproximinal sets in some Banach spaces”, Math. Balkanica, 4 (1974), 79–82 | MR | Zbl

[4] Cobzaş S., “Antiproximinal sets in Banach spaces of continuous function”, Revue d'Analyse Numérique et de la Théorie de l'Approximation, 5:2 (1976), 127–143 | Zbl

[5] Cobzaş S., “Antiproximinal sets in Banach spaces of $c_0$-type”, Revue d'Analyse Numérique et de la Théorie de l'Approximation, 7:2 (1978), 141–145 | Zbl

[6] Fonf V. P., “Ob antiproksiminalnykh mnozhestvakh v prostranstvakh nepreryvnykh funktsii na bikompaktakh”, Matem. zametki, 33:3 (1983), 549–558 | MR | Zbl

[7] Singer I., Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, Berlin–N. Y., 1970

[8] Danford N., Shvarts Dzh., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[9] Shefer Kh., Topologicheskie vektornye prostranstva, Mir, M., 1971

[10] Schaefer H. H., Banach Lattices and Positive Operator, Dis. Grundlehren Wissenschaften in Einzeldarstellunden, 215, Springer Verlag, Berlin–N. Y., 1974 | Zbl

[11] Kudryavtsev L. D., “Ob ekvivalentnykh normakh v vesovykh prostranstvakh”, Tr. MIAN, 170, Nauka, M., 1984, 161–190 | MR | Zbl

[12] Phelps R. R., “Counterexamples concerning support theorems for convex sets in Hilbert space”, Canad. Math. Bull., 31:1 (1988), 121–128 | MR | Zbl

[13] Fonf V. P., “O silno antiproksiminalnykh mnozhestvakh v banakhovykh prostranstvakh”, Matem. zametki, 47:2 (1990), 130–136 | MR

[14] Semadeni Z., Banach spaces of continuous functions, Polish scientific publishers, Warszawa, 1971 | Zbl

[15] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961

[16] Bourgain J., “On dentability and the Bishop–Phelps property”, Israel J. Math., 28:4 (1977), 265–271 | DOI | MR | Zbl

[17] Distel Dzh., Geometriya banakhovykh prostranstv, Vischa shkola, Kiev, 1980

[18] Edelstien M., “A note on nearest points”, Quart. J. Math., 21:84 (1970), 403–406 | DOI