Geodesic
Uniformly and locally convex asymmetric spaces
Sbornik. Mathematics, Tome 213 (2022) no. 10, pp. 1444-1469 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The nonemptyness of the intersections of nested systems of convex bounded closed subsets of uniformly convex asymmetric spaces is studied. The density properties of the points of existence and points of approximative uniqueness are examined for nonempty closed subsets of uniformly convex asymmetric spaces. Problems of the existence and stability of Chebyshev centres are considered; the relationships between $\gamma$-suns, suns and the existence of best approximants are investigated. Sufficient conditions for radial $\delta$-solarity are obtained. Bibliography: 27 titles.
Keywords: asymmetric space, uniformly convex space, Chebyshev centre, approximative uniqueness, convex set, sun. 
@article{SM_2022_213_10_a5,
     author = {I. G. Tsar'kov},
     title = {Uniformly and locally convex asymmetric spaces},
     journal = {Sbornik. Mathematics},
     pages = {1444--1469},
     year = {2022},
     volume = {213},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2022_213_10_a5/}
}
TY  - JOUR
AU  - I. G. Tsar'kov
TI  - Uniformly and locally convex asymmetric spaces
JO  - Sbornik. Mathematics
PY  - 2022
SP  - 1444
EP  - 1469
VL  - 213
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2022_213_10_a5/
LA  - en
ID  - SM_2022_213_10_a5
ER  - 
%0 Journal Article
%A I. G. Tsar'kov
%T Uniformly and locally convex asymmetric spaces
%J Sbornik. Mathematics
%D 2022
%P 1444-1469
%V 213
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2022_213_10_a5/
%G en
%F SM_2022_213_10_a5
I. G. Tsar'kov. Uniformly and locally convex asymmetric spaces. Sbornik. Mathematics, Tome 213 (2022) no. 10, pp. 1444-1469. http://geodesic.mathdoc.fr/item/SM_2022_213_10_a5/

[1] Ş. Cobzaş, Functional analysis in asymmetric normed spaces, Front. Math., Birkhäuser/Springer Basel AG, Basel, 2013, x+219 pp. | DOI | MR | Zbl

[2] V. Donjuán and N. Jonard-Pérez, “Separation axioms and covering dimension of asymmetric normed spaces”, Quaest. Math., 43:4 (2020), 467–491 | DOI | MR | Zbl

[3] S. Cobzaş, “Separation of convex sets and best approximation in spaces with asymmetric norm”, Quaest. Math., 27:3 (2004), 275–296 | DOI | MR | Zbl

[4] A. R. Alimov, “The Banach-Mazur theorem for spaces with an asymmetric distance”, Uspekhi Mat. Nauk, 58:2(350) (2003), 159–160 ; English transl. in Russian Math. Surveys, 58:2 (2003), 367–369 | DOI | MR | Zbl | DOI

[5] A. R. Alimov, “On the structure of the complements of Chebyshev sets”, Funktsional. Anal. i Prilozhen., 35:3 (2001), 19–27 ; English transl. in Funct. Anal. Appl., 35:3 (2001), 176–182 | DOI | MR | Zbl | DOI

[6] A. R. Alimov, “Convexity of bounded Chebyshev sets in finite-dimensional spaces with asymmetric norm”, Izv. Saratov. Univ. (N.S.) Ser. Mat. Mekh. Informat., 14:4(2) (2014), 489–497 (Russian) | DOI | Zbl

[7] I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Mat. Sb., 211:8 (2020), 132–157 ; English transl. in Sb. Math., 211:8 (2020), 1190–1211 | DOI | MR | Zbl | DOI

[8] I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Mat. Sb., 210:9 (2019), 129–152 ; English transl. in Sb. Math., 210:9 (2019), 1326–1347 | DOI | MR | Zbl | DOI

[9] I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Ross. Akad. Nauk Ser. Mat., 82:4 (2018), 199–224 ; English transl. in Izv. Math., 82:4 (2018), 837–859 | DOI | MR | Zbl | DOI

[10] I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Mat. Sb., 209:4 (2018), 95–116 ; English transl. in Sb. Math., 209:4 (2018), 560–579 | DOI | MR | Zbl | DOI

[11] I. G. Tsar'kov, “Properties of monotone path-connected sets”, Izv. Ross. Akad. Nauk. Ser. Mat., 85:2 (2021), 142–171 ; English transl. in Izv. Math., 85:2 (2021), 306–331 | DOI | MR | Zbl | DOI

[12] I. G. Tsar'kov, “Uniform convexity in nonsymmetric spaces”, Mat. Zametki, 110:5 (2021), 773–785 ; English transl. in Math. Notes, 110:5 (2021), 773–783 | DOI | MR | Zbl | DOI

[13] G. G. Magaril-Il'yaev and V. M. Tikhomirov, Convex analysis: theory and applications, 3d revised ed., Librokom, Moscow, 2011, 176 pp.; English transl. of 2nd ed., Transl. Math. Monogr., 222, Amer. Math. Soc., Providence, RI, 2003, viii+183 pp. | MR | Zbl

[14] M. G. Kreĭn, “The $L$-problem in an abstract linear normed space”, Some questions in the theory of moments, GONTI, Kharkov, 1938, 171–199; English transl., Transl. Math. Monogr., 2, Amer. Math. Soc., Providence, RI, 1962, 175–204 | MR | Zbl

[15] H. König, “Sublineare Funktionale”, Arch. Math. (Basel), 23 (1972), 500–508 | DOI | MR | Zbl

[16] H. König, “Sublinear functionals and conical measures”, Arch. Math. (Basel), 77:1 (2001), 56–64 | DOI | MR | Zbl

[17] V. F. Babenko, “Nonsymmetric approximation in spaces of integrable functions”, Ukrain. Mat. Zh., 34:4 (1982), 409–416 ; English transl. in Ukrainian Math. J., 34:4 (1982), 331–336 | MR | Zbl | DOI

[18] E. P. Dolzhenko and E. A. Sevast'yanov, “Approximations with a sign-sensitive weight: existence and uniqueness theorems”, Izv. Ross. Akad. Nauk Ser. Mat., 62:6 (1998), 59–102 ; English transl. in Izv. Math., 62:6 (1998), 1127–1168 | DOI | MR | Zbl | DOI

[19] E. P. Dolzhenko and E. A. Sevast'yanov, “Approximations with a sign-sensitive weight. Stability, applications to the theory of snakes and Hausdorff approximations”, Izv. Ross. Akad. Nauk Ser. Mat., 63:3 (1999), 77–118 ; English transl. in Izv. Math., 63:3 (1999), 495–534 | DOI | MR | Zbl | DOI

[20] L. Collatz and W. Krabs, Approximationstheorie. Tschebyscheffsche Approximation mit Anwendungen, B. G. Teubner, Stuttgart, 1973, 208 pp. | MR | Zbl

[21] R. C. Flagg and R. D. Kopperman, “The asymmetric topology of computer science”, Mathematical foundations of programming semantics (New Orleans, LA 1993), Lecture Notes in Comput. Sci., 802, Springer, Berlin, 1993, 544–553 | DOI | MR | Zbl

[22] P. A. Borodin, “Quasiorthogonal sets and conditions for a Banach space to be a Hilbert space”, Mat. Sb., 188:8 (1997), 63–74 ; English transl. in Sb. Math., 188:8 (1997), 1171–1182 | DOI | MR | Zbl | DOI

[23] P. A. Borodin, “The Banach-Mazur theorem for spaces with asymmetric norm”, Mat. Zametki, 69:3 (2001), 329–337 ; English transl. in Math. Notes, 69:3 (2001), 298–305 | DOI | MR | Zbl | DOI

[24] G. E. Ivanov and M. S. Lopushanski, “Well-posedness of approximation and optimization problems for weakly convex sets and functions”, Fundam. Prikl. Mat., 18:5 (2013), 89–118 ; English transl. in J. Math. Sci. (N.Y.), 209:1 (2015), 66–87 | MR | Zbl | DOI

[25] G. E. Ivanov and M. S. Lopushanski, “Approximation properties of weakly convex sets in spaces with asymmetric seminorm”, Tr. Moskov. Fiz. Tekhn. Inst., 4:4 (2012), 94–104 (Russian)

[26] A. R. Alimov and I. G. Tsar'kov, “Chebyshev centres, Jung constants, and their applications”, Uspekhi Mat. Nauk, 74:5(449) (2019), 3–82 ; English transl. in Russian Math. Surveys, 74:5 (2019), 775–849 | DOI | MR | Zbl | DOI

[27] A. R. Alimov and I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Uspekhi Mat. Nauk, 71:1(427) (2016), 3–84 ; English transl. in Russian Math. Surveys, 71:1 (2016), 1–77 | DOI | MR | Zbl | DOI