Geodesic
The Lipschitz constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness
Izvestiya. Mathematics , Tome 62 (1998) no. 2, pp. 313-318.

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

We obtain upper estimates for the Lipschitz constants of the metric $\varepsilon$-projection operator $P$ in terms of the modules of convexity and smoothness of the space when the following three parameters are varied: the approximee $x$, the convex approximating set $M$, and the accuracy of approximation $\varepsilon>0$. These estimates are unimprovable in the class of all normed linear spaces. We use them to obtain new stability evaluations for continuous selectors of the operator $P$. 
@article{IM2_1998_62_2_a4,
     author = {A. V. Marinov},
     title = {The {Lipschitz} constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness},
     journal = {Izvestiya. Mathematics },
     pages = {313--318},
     publisher = {mathdoc},
     volume = {62},
     number = {2},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1998_62_2_a4/}
}
TY  - JOUR
AU  - A. V. Marinov
TI  - The Lipschitz constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness
JO  - Izvestiya. Mathematics 
PY  - 1998
SP  - 313
EP  - 318
VL  - 62
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1998_62_2_a4/
LA  - en
ID  - IM2_1998_62_2_a4
ER  - 
%0 Journal Article
%A A. V. Marinov
%T The Lipschitz constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness
%J Izvestiya. Mathematics 
%D 1998
%P 313-318
%V 62
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1998_62_2_a4/
%G en
%F IM2_1998_62_2_a4
A. V. Marinov. The Lipschitz constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness. Izvestiya. Mathematics , Tome 62 (1998) no. 2, pp. 313-318. http://geodesic.mathdoc.fr/item/IM2_1998_62_2_a4/

[1] Marinov A. V., “Ustoichivost $\varepsilon $-kvazireshenii operatornykh uravnenii I roda”, Priblizhenie funktsii polinomami i splainami, Sb. st., UNTs AN SSSR, Sverdlovsk, 1985, 105–117 | MR

[2] Marinov A. V., “Otsenki ustoichivosti metricheskoi $\varepsilon $-proektsii cherez modul vypuklosti prostranstva”, Tr. IMM UrO RAN, 2 (1992), 85–109 | MR | Zbl

[3] Wulbert D. E., Continuity of metric projections. Approximation theory in a normed linear lattice, Thes. Univ. Texas Comp. Center, Austin, 1966, P. 105

[4] Liskovets O. A., “Metod $\varepsilon $-kvazireshenii dlya uravnenii I roda”, Differents. uravn., 9:10 (1973), 1851–1861 | MR | Zbl

[5] Liskovets O. A., Variatsionnye metody resheniya neustoichivykh zadach, Nauka i tekhnika, Minsk, 1981 | MR | Zbl

[6] Berdyshev V. I., “Varirovanie normy v zadache o nailuchshem priblizhenii”, Matem. zametki, 29:2 (1981), 181–196 | MR | Zbl

[7] Berdyshev V. I., “Nepreryvnost mnogoznachnogo otobrazheniya, svyazannogo s zadachei minimizatsii funktsionala”, Izv. AN SSSR. Ser. matem., 44:3 (1980), 483–509 | MR | Zbl

[8] Marinov A. V., “Nepreryvnost i svyaznost metricheskoi $\delta $-proektsii”, Approksimatsiya v konkretnykh i abstraktnykh banakhovykh prostranstvakh, Sb. st., UNTs AN SSSR, Sverdlovsk, 1987, 82–95 | MR

[9] Attouch H., Wets R. J.-B., Lipschitsian stability of $\varepsilon $-approximate solutions in convex optimisation, WP–87–25, IIASA, Laxenburg, 1987, 31 pp.

[10] Attouch H., Wets R. J.-B., Quantitative stability of variational systems. I: The epigraphical distance, WP–88–8, IIASA, Laxenburg, 1988, 41 pp.

[11] Garkavi A. L., “Teoriya nailuchshego priblizheniya v lineinykh normirovannykh prostranstvakh”, Itogi nauki i tekhniki. Matem. analiz, VINITI, M., 1969, 75–132

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

[13] Deutsch F., “A survey of metric selections”, Contemporery Math., 18 (1983), 49–71 | MR | Zbl

[14] Figiel T., “On the moduli of convexity and smoothness”, Studia Math., LVI:2 (1976), 121–155 | MR

[15] Gurarii V. I., “O differentsialnykh svoistvakh modulei vypuklosti banakhovykh prostranstv”, Matematicheskie issledovaniya, T. II, no. 1, RIO AN MSSR, Kishinev, 1966, 141–148 | MR

[16] Distel Dzh., Geometriya banakhovykh prostranstv, Vischa shk., Kiev, 1980 | MR

[17] Björnestål B. O., “Local Lipschitz continuity of the metric projection operator”, Banach Center Publications, 4 (1979), 43–54 | MR

[18] Kadets M. I., “O topologicheskoi ekvivalentnosti ravnomerno vypuklykh prostranstv”, UMN, 10:4 (1955), 137–141 | MR | Zbl

[19] Deutsch F., Li W., Park S.-H., “Characterization of continuous and Lipschitz continuous metric selections in normed linear spaces”, J. Approxim. Theory, 58:3 (1989), 297–314 | DOI | MR | Zbl

[20] Brown A. L., “Set valued mappings, continuous selections, and metric projections”, J. Approxim. Theory, 57:1 (1989), 48–68 | DOI | MR | Zbl

[21] Konyagin S. V., “O nepreryvnykh operatorakh obobschennogo ratsionalnogo priblizheniya”, Matem. zametki, 44:3 (1988), 404 | MR | Zbl

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

[23] Albrekht P. V., “Poryadki modulei nepreryvnosti operatorov pochti nailuchshego priblizheniya”, Matem. sb., 185:9 (1994), 3–28

[24] Marinov A. V., “Otsenki ustoichivosti nepreryvnoi selektsii dlya metricheskoi pochti-proektsii”, Matem. zametki, 55:4 (1994), 47–53 | MR | Zbl

[25] Michael E., “Continuous selections”, J. Ann. Math. Ser. 2, 63:2 (1956), 361–381 | DOI | MR

[26] Gryunbaum B., Etyudy po kombinatornoi geometrii i teorii vypuklykh tel, Nauka, M., 1971 | MR | Zbl

[27] Positselskii E. D., “O lipshitsevykh otobrazheniyakh v prostranstve vypuklykh tel”, Optimizatsiya, Sb. trudov, no. 4 (21), 1971, 83–89

[28] Shepard G. C., “The Steiner point of a convex polytope”, Canad. J. Math., 18:6 (1966), 1294–1300 | MR