The $n$-antiproximinal sets

B. B. Bednov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2015), pp. 29-34

Résumé

The notion of $n$-antiproximinal set in a Banach space is defined. The existence of convex closed $n$-antiproximinal sets in the spaces $C$ and $L_1$ is studied.

Bibliographie

[1] Holmes R.B., A course on optimization and best approximation, Lect. Notes Math., 257, Springer-Verlag, Berlin, 1972 | DOI | MR | Zbl

[2] Klee V., “Remarks on nearest points in normed linear spaces”, Proc. Colloq. Convexity (Copenhagen 1965), Copenhagen, 1967, 161–176 | MR

[3] Singer I., Best approximation in normed linear spaces by elements of linear subspaces, Editura Academiei, Bucharest; Springer-Verlag, Berlin, 1970 | MR

[4] Edelstein M., “A note on nearest points”, Quart. J. Math., 21 (1970), 403–407 | DOI | MR

[5] Edelstein M., Thompson A.C., “Some results on nearest points and support properties of convex sets in $\mathrm{ \bf c_0}$”, Pacif. J. Math., 40:3 (1972), 553–560 | DOI | MR | Zbl

[6] Kobzash S., “Vypuklye antiproksiminalnye mnozhestva v prostranstvakh $\mathrm{ \bf c_0}$ i $\mathrm{ \bf c}$”, Matem. zametki, 17 (1975), 449–457 | Zbl

[7] Cobzaş S., “Antiproximinal sets in Banach spaces of continuous functions”, Rev. anal. numer. şi teor. aproxim., 5 (1976), 127–143 | MR | Zbl

[8] Cobzaş S., “Antiproximinal sets in Banach spaces of $c_0$-type”, Rev. anal. numer. şi teor. aproxim., 7 (1978), 141–145 | MR | Zbl

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

[10] Borwein J.M., “Some remarks on a paper of S. Cobzas on antiproximinal sets”, Bull. Calcutta Math. Soc., 73 (1981), 5–8 | MR | Zbl

[11] Edelstein M., “Weakly proximinal sets”, J. Approxim. Theory, 18:1 (1976), 1–8 | DOI | MR | Zbl

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

[13] Floret K., “On the sum of two closed convex sets”, Meth. Oper. Res., 36 (1978), 73–85 | MR

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

[15] Balaganskii V.S., “Antiproksiminalnye mnozhestva v prostranstvakh nepreryvnykh funktsii”, Matem. zametki, 60:5 (1996), 643–657 | DOI | MR

[16] Balaganskii V.S., “Ob antiproksiminalnykh mnozhestvakh v prostranstve Grotendika”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18, no. 4, 2012, 90–103

[17] Borwein J.M., Jiménez-Sevilla M., Moreno J.P., “Antiproximinal norms in Banach spaces”, J. Approxim. Theory, 114 (2002), 57–69 | DOI | MR | Zbl

[18] Cobzaş S., “Antiproximinal sets in Banach spaces”, Acta Univ. carol. math. et phys., 40:2 (1999), 43–52 | MR | Zbl

[19] Borodin P.A., “O vypuklosti $N$-chebyshevskikh mnozhestv”, Izv. RAN. Ser. matem., 75:5 (2011), 19–46 | DOI | MR | Zbl

[20] Rubinshtein G.Sh., “Ob odnoi ekstremalnoi zadache v lineinom normirovannom prostranstve”, Sib. matem. zhurn., VI:3 (1965), 711–714 | Zbl

[21] Distel Dzh., Geometriya banakhovykh prostranstv, Vischa shkola, Kiev, 1980 | MR

[22] Danford N., Shvarts Dzh., Lineinye operatory. Obschaya teoriya, IL, M., 1962 | MR

[23] Bednov B.B., “O tochkakh Shteinera v prostranstve nepreryvnykh funktsii”, Vestn. Mosk. un-ta. Matem. Mekhan., 2011, no. 6, 26–31 | MR | Zbl