Geodesic
Properties of the distance function to strongly and weakly convex sets in a nonsymmetrical space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2020), pp. 22-38.

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

We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly and weakly convex sets in an arbitrary Hilbert space. Some properties of the caliber of a strongly convex set and the conditions for obtaining a strict, strong or weak convexity of Lebesgue sets of the distance function are established in accordance with the requirements for the set, the caliber of which specifies the distance function, and the set to which the distance is measured. The corresponding inequalities are obtained that reflect the behavior of the distance function on segments and allow comparing it with strictly, strongly or weakly convex functions.
Keywords: gauge of a set, distance function (DF), strongly and weakly convex sets and functions. 
@article{IVM_2020_5_a2,
     author = {S. I. Dudov and E. S. Polovinkin and V. V. Abramova},
     title = {Properties of the distance function to strongly and weakly convex sets in a nonsymmetrical space},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {22--38},
     publisher = {mathdoc},
     number = {5},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2020_5_a2/}
}
TY  - JOUR
AU  - S. I. Dudov
AU  - E. S. Polovinkin
AU  - V. V. Abramova
TI  - Properties of the distance function to strongly and weakly convex sets in a nonsymmetrical space
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2020
SP  - 22
EP  - 38
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2020_5_a2/
LA  - ru
ID  - IVM_2020_5_a2
ER  - 
%0 Journal Article
%A S. I. Dudov
%A E. S. Polovinkin
%A V. V. Abramova
%T Properties of the distance function to strongly and weakly convex sets in a nonsymmetrical space
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2020
%P 22-38
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2020_5_a2/
%G ru
%F IVM_2020_5_a2
S. I. Dudov; E. S. Polovinkin; V. V. Abramova. Properties of the distance function to strongly and weakly convex sets in a nonsymmetrical space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2020), pp. 22-38. http://geodesic.mathdoc.fr/item/IVM_2020_5_a2/

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

[2] Pshenichnyi B. N., Vypuklyi analiz i ekstremalnye zadachi, Nauka, M., 1980 | MR

[3] Dunham Ch.B., “Asymmetric norms and linear approximation”, Congr. Numer., 69 (1989), 113–120 | MR | Zbl

[4] Romaguera S., Schellekens M., “Quasi-metric properties of complexity spaces”, Topology Appl., 98:1 (1999), 311–322 | DOI | MR | Zbl

[5] De Blasi F. S., Myjak J., “On a generalized best approximation problem”, J. Approx.Theory, 94:1 (1998), 54–72 | DOI | MR | Zbl

[6] Alegre C., “Continuous operators on asymmetric normed spaces”, Acta Math. Hung., 122:4 (2009), 357–372 | DOI | MR | Zbl

[7] Cobzas S., Functional analysis in asymmetric normed spaces, Birkhauser, Basel, 2013 | MR | Zbl

[8] Alimov A. R., Approksimativno-geometricheskie svoistva mnozhestv v normirovannykh i nesimmetrichno normirovannykh prostranstvakh, Diss. na soiskanie uchenoi stepeni doktora fiz.-matem. nauk, Moskovsk. gos. un-ta, 2014

[9] Alimov A. R., “Vypuklost ogranichennykh chebyshevskikh mnozhestv v konechnomernykh prostranstvakh s nesimmetrichnoi normoi”, Izv. Saratovsk. un-ta, Novaya seriya. Ser. Matem. Mekhan. Informatika, 16:2 (2016), 133–137 | MR | Zbl

[10] Ivanov G. E., Lopushanski M. S., “Approksimativnye svoistva slabo vypuklykh mnozhestvo v prostranstvakh s nesimmetrichnoi polunormoi”, Tr. MFTI, 4:4 (2012), 94–104

[11] Ivanov G. E., Lopushanski M. S., “Separation theorems for nonconvex sets in space with nonsymmetric seminorm”, J. Math. Inequalities and Appl., 20:3 (2017), 737–754 | MR | Zbl

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

[13] Dudov S. I., Zlatorunskaya I. V., “Ravnomernaya otsenka vypuklogo kompakta sharom proizvolnoi normy”, Matem. sb., 191:10 (2000), 13–38 | DOI | MR | Zbl

[14] Vasilev F. P., Metody optimizatsii, v. I, MTsNMO, M., 2011

[15] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990

[16] Vial J.-P., “Strong and weak convexity of sets and functions”, Math. Oper. Res., 8:2 (1983), 231–259 | DOI | MR | Zbl

[17] Ivanov G. E., Slabo vypuklye mnozhestva i funktsii: teoriya i prilozheniya, Fizmatlit, M., 2006

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

[19] Golubev M. O., “Svyaz silno vypuklykh mnozhestv s silnoi vypuklostyu funktsii rasstoyaniya i slaboi vognutostyu funktsii antirasstoyaniya”, Sovr. probl. teorii funktsii i ikh prilozheniya, Mater. 17-i mezhdunarodn. Saratovsk. zimn. shk., Izd-vo Nauchn. kn., Saratov, 2014, 76–78