Geodesic
Locally Chebyshev Sets on the Plane
Matematičeskie zametki, Tome 97 (2015) no. 1, pp. 142-149.

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

The notion of locally Chebyshev set is introduced and the relationship between Chebyshev and locally Chebyshev sets on the plane is studied. It is shown that any connected closed locally Chebyshev set in an arbitrary Banach space is a Chebyshev set and each Chebyshev set is locally Chebyshev if and only if the space is strictly convex.
Keywords: Chebyshev set, locally Chebyshev set, metric projection, Banach space. 
@article{MZM_2015_97_1_a13,
     author = {A. A. Flerov},
     title = {Locally {Chebyshev} {Sets} on the {Plane}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {142--149},
     publisher = {mathdoc},
     volume = {97},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_97_1_a13/}
}
TY  - JOUR
AU  - A. A. Flerov
TI  - Locally Chebyshev Sets on the Plane
JO  - Matematičeskie zametki
PY  - 2015
SP  - 142
EP  - 149
VL  - 97
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2015_97_1_a13/
LA  - ru
ID  - MZM_2015_97_1_a13
ER  - 
%0 Journal Article
%A A. A. Flerov
%T Locally Chebyshev Sets on the Plane
%J Matematičeskie zametki
%D 2015
%P 142-149
%V 97
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2015_97_1_a13/
%G ru
%F MZM_2015_97_1_a13
A. A. Flerov. Locally Chebyshev Sets on the Plane. Matematičeskie zametki, Tome 97 (2015) no. 1, pp. 142-149. http://geodesic.mathdoc.fr/item/MZM_2015_97_1_a13/

[1] N. V. Efimov, S. B. Stechkin, “Nekotorye svoistva chebyshevskikh mnozhestv”, Dokl. AN SSSR, 118:1 (1958), 17–19 | MR | Zbl

[2] E. S. Polovinkin, M. V. Balashov, Elementy vypuklogo i silno vypuklogo analiza, Fizmatlit, M., 2007 | Zbl

[3] A. V. Arutyunov, “Vypuklye svoistva preobrazovaniya Lezhandra”, Matem. zametki, 28:2 (1980), 255–264 | MR | Zbl

[4] L. P. Vlasov, “Approksimativnye svoistva mnozhestv v lineinykh normirovannykh prostranstvakh”, UMN, 28:6 (1973), 3–66 | MR | Zbl

[5] E. Asplund, B. Grünbaum, “On the geometry of Minkowski planes”, Enseignement Math. (2), 6 (1960), 299–306 | MR

[6] L. N. H. Bunt, Bijdrage tot de theorie der convexe puntverzamelingen, Thesis, Univ. Groningen, Amsterdam, 1934

[7] T. Motzkin, “Sur quelques propriétés caractéristiques des ensembles bornés non convexes”, Rend. Accad. Naz. Lincei, 21 (1935), 773–779 | Zbl

[8] N. V. Efimov, S. B. Stechkin, “Opornye svoistva mnozhestv v banakhovykh prostranstvakh i chebyshevskie mnozhestva”, Dokl. AN SSSR, 127:2 (1959), 254–257 | MR | Zbl