Geodesic
Strongly convex analysis
Sbornik. Mathematics, Tome 187 (1996) no. 2, pp. 259-286 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Properties of strongly convex sets (that is, of sets that can be represented as intersections of balls of radius fixed for each particular set) are investigated. A connection between strongly convex sets and strongly convex functions is established. The concept of a strongly convex $R$-hull of a set (the minimal strongly convex set containing the given set) is introduced; an explicit formula for the srongly convex $R$-hull of a set is obtained. The behaviour of the strongly convex $R$-hull under the variation of $R$ and of the sets is considered. An analogue of the Carathéodory theorem for strongly convex sets is obtained. The concept of a strongly extreme point is introduced, and a generalization of the Krein–Mil'man theorem for strongly convex sets is proved. Polyhedral approximations of convex and, in particular, of strongly convex compact sets are considered. Sharp error estimates for polyhedral and strongly convex approximations of such sets from inside and outside are established. 
@article{SM_1996_187_2_a5,
     author = {E. S. Polovinkin},
     title = {Strongly convex analysis},
     journal = {Sbornik. Mathematics},
     pages = {259--286},
     year = {1996},
     volume = {187},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_2_a5/}
}
TY  - JOUR
AU  - E. S. Polovinkin
TI  - Strongly convex analysis
JO  - Sbornik. Mathematics
PY  - 1996
SP  - 259
EP  - 286
VL  - 187
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1996_187_2_a5/
LA  - en
ID  - SM_1996_187_2_a5
ER  - 
%0 Journal Article
%A E. S. Polovinkin
%T Strongly convex analysis
%J Sbornik. Mathematics
%D 1996
%P 259-286
%V 187
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1996_187_2_a5/
%G en
%F SM_1996_187_2_a5
E. S. Polovinkin. Strongly convex analysis. Sbornik. Mathematics, Tome 187 (1996) no. 2, pp. 259-286. http://geodesic.mathdoc.fr/item/SM_1996_187_2_a5/

[1] Pliś A., “Accessible Sets in Control Theory”, Int. Conf. on Diff. Eqs., Academic Press, 1975, 646–650 | MR

[2] Lojasiewicz St.\rom(Jr.\rom), “Some properties of accessible sets in non linear control systems”, Ann. Polon. Math., XXXVII:2 (1979), 123–127 | MR

[3] Frankowska H., Olech C., “$R$-convexity of the Integral of the Set-Valued Functions”, Contributions to Analysis and Geometry, Johns Hopkins Univ. Press, Baltimore, Md., 1981, 117–129 | MR

[4] Krasnoselskii M. A., Pokrovskii A. V., Sistemy s gisterezisom, Nauka, M., 1983 | MR

[5] Pontryagin L. S., “Lineinye differentsialnye igry presledovaniya”, Matem. sb., 112:3 (1980), 307–330 | MR | Zbl

[6] Rokafellar R., Vypuklyi analiz, Mir, M., 1973

[7] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR | Zbl

[8] Polyak B. T., Vvedenie v optimizatsiyu, Nauka, M., 1983 | MR

[9] Polovinkin E. S., Elementy teorii mnogoznachnykh otobrazhenii, Ucheb. posobie, Izd-vo MFTI, M., 1982

[10] Polovinkin E. S., “Ob integrirovanii mnogoznachnykh otobrazhenii”, DAN SSSR, 271:5 (1983), 1069–1074 | MR | Zbl

[11] Polovinkin E. S., “O svoistvakh silno vypuklykh mnozhestv”, Modelirovanie protsessov upravleniya i obrabotki informatsii, Izd-vo MFTI, M., 1994, 182–189

[12] Polovinkin E. S., “O vypuklykh i silno vypuklykh approksimatsiyakh mnozhestv”, DAN, 1995 (to appear)

[13] Berger M., Geometrie, CEDIC/FERNAND NATHAN, Paris, 1977

[14] Ivanov G. E., Polovinkin E. S., “Vtoroi poryadok skhodimosti algoritma vychisleniya tseny lineinoi differentsialnoi igry”, DAN, 340:2 (1995), 151–154 | MR | Zbl

[15] Ivanov G. E., Polovinkin E. S., “O silno vypuklykh lineinykh differentsialnykh igrakh”, Differents. uravneniya, 31:10 (1995)

[16] Rudin U., Funktsionalnyi analiz, Mir, M., 1975 | MR

[17] Polovinkin E. S., “On Strongly Convex Sets”, Phystech. Journal, 2:1 (1995) | MR | Zbl

[18] Polovinkin E. S., “Obobschenie teorem Karateodori i Kreina–Milmana dlya silno vypuklykh mnozhestv”, DAN, 1995 (to appear)