Geodesic
Properties of Sets Admitting Stable $\varepsilon$-Selections
Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 608-613.

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

Sets in which some convex subsets admit local (global) continuous $\varepsilon$-selections are studied. In particular, it is shown that if, for any number $\varepsilon>0$, some neighborhood $O(x)$ of a point $x$ in a Banach space $X$ contains a dense (in $O(x)$) convex set $K$ admitting an upper semicontinuous acyclic (in particular, continuous single-valued) $\varepsilon$-selection to an approximatively compact set $M\subset X$, then $x$ is a $\delta$-sun point; if, in addition, $X\in (R)$, then the set of all points nearest to $x$ in $M$ is a singleton.
Keywords: Banach space, acyclic upper semicontinuous $\varepsilon$-selection, approximatively compact set, $\delta$-sun, convex set. 
@article{MZM_2011_89_4_a12,
     author = {I. G. Tsar'kov},
     title = {Properties of {Sets} {Admitting} {Stable} $\varepsilon${-Selections}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {608--613},
     publisher = {mathdoc},
     volume = {89},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a12/}
}
TY  - JOUR
AU  - I. G. Tsar'kov
TI  - Properties of Sets Admitting Stable $\varepsilon$-Selections
JO  - Matematičeskie zametki
PY  - 2011
SP  - 608
EP  - 613
VL  - 89
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a12/
LA  - ru
ID  - MZM_2011_89_4_a12
ER  - 
%0 Journal Article
%A I. G. Tsar'kov
%T Properties of Sets Admitting Stable $\varepsilon$-Selections
%J Matematičeskie zametki
%D 2011
%P 608-613
%V 89
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a12/
%G ru
%F MZM_2011_89_4_a12
I. G. Tsar'kov. Properties of Sets Admitting Stable $\varepsilon$-Selections. Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 608-613. http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a12/

[1] I. G. Tsarkov, “O svyaznosti nekotorykh klassov mnozhestv v banakhovykh prostranstvakh”, Matem. zametki, 40:2 (1986), 174–196 | MR | Zbl

[2] N. Stinrod, S. Eilenberg, Osnovaniya algebraicheskoi topologii, Fizmatgiz, M., 1958 | MR | Zbl

[3] I. G. Tsarkov, “Svoistva mnozhestv, obladayuschikh nepreryvnoi vyborkoi iz operatora $P^\delta$”, Matem. zametki, 48:4 (1990), 122–131 | MR | Zbl

[4] V. S. Balaganskii, L. P. Vlasov, “Problema vypuklosti chebyshëvskikh mnozhestv”, UMN, 51:6 (1996), 125–188 | MR | Zbl

[5] L. P. Vlasov, “Approksimativnye svoistva mnozhestv v banakhovykh prostranstvakh”, Matem. zametki, 7:5 (1970), 594–604 | MR | Zbl

[6] L. P. Vlasov, “Nekotorye teoremy o chebyshevskikh mnozhestvakh”, Matem. zametki, 11:2 (1972), 135–144 | MR | Zbl