On finite-dimensional Banach spaces in which suns are connected. (English) Zbl 1474.46027
Summary: The present paper extends and refines some results on the connectedness of suns in finite-dimensional normed linear spaces. In particular, a sun in a finite-dimensional \((BM)\)-space is shown to be monotone path-connected and having a continuous multiplicative (additive) \( \varepsilon \)-selection from the operator of nearly best approximation for any \(\varepsilon>0\). New properties of \((BM)\)-space are put forward.
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 41A50 | Best approximation, Chebyshev systems |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
Keywords:
sun; strict sun; bounded connectedness; \((BM)\)-space; contractibility; nearly best approximation; \( \varepsilon \)-selection; Menger connectedness; monotone path-connectednessReferences:
| [1] | A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces”, Izvestiya: Mathematics, 78:4 (2014), 641-655 · Zbl 1303.41018 · doi:10.1070/IM2014v078n04ABEH002702 |
| [2] | A. R. Alimov, “The Rainwater-Simons weak convergence theorem for the Brown associated norm”, Eurasian Math. J., 5:2 (2014), 126-131 · Zbl 1326.46006 |
| [3] | A. R. Alimov, “Monotone path-connectedness of \(R\)-weakly convex sets in spaces with linear ball embedding”, Eurasian Math. J., 3:2 (2012), 21-30 · Zbl 1269.46009 |
| [4] | A. R. Alimov, “Connectedness of suns in the space \(c_0\)”, Izvestiya: Mathematics, 69:4 (2005), 3-18 · Zbl 1095.46009 · doi:10.1070/IM2005v069n04ABEH001646 |
| [5] | A. R. Alimov, “Convexity of Chebyshev sets contained in a subspace”, Math. Notes, 78:1 (2005), 3-15 · Zbl 1095.46008 · doi:10.1007/s11006-005-0093-0 |
| [6] | A. R. Alimov, “Local solarity of suns in normed linear spaces”, Fundam. Prikl. Mat., 17:7 (2011/2012), 3-14 |
| [7] | A. R. Alimov, “Monotone path-connectedness of Chebyshev sets in the space \(C(Q)\)”, Sbornik: Mathematics, 197:9 (2006), 3-18 · Zbl 1147.41011 · doi:10.4213/sm1129 |
| [8] | A. R. Alimov, V. Yu. Protasov, “Separation of convex sets by extreme hyperplanes”, Fundam. Prikl. Mat., 17:4 (2011/2012), 3-12 · Zbl 1280.52022 |
| [9] | A. R. Alimov, I. G. Tsar’kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, Fundam. Prikl. Matem., 19:4 (2014), 21-91 |
| [10] | B. B. Bednov, P. A. Borodin, “Banach spaces that realize minimal fillings”, Sbornik: Mathematics, 205:4 (2014), 459-475 · Zbl 1327.46013 · doi:10.1070/SM2014v205n04ABEH004383 |
| [11] | H. Berens, L. Hetzelt, “Die metrische Struktur der Sonnen in \(l^\infty(n)\)”, Aequat. Math., 27 (1984), 274-287 · Zbl 0544.41025 · doi:10.1007/BF02192677 |
| [12] | V. Boltyanski, H. Martini, P. S. Soltan, Excursions into combinatorial geometry, Springer, Berlin, 1997 · Zbl 0877.52001 |
| [13] | A. L. Brown, “Suns in polyhedral spaces”, Seminar of Mathem. Analysis, Proceedings (Univ. Malaga and Seville, Spain, Sept. 2002-Feb. 2003), eds. D. G. Álvarez, G. Lopez Acedo, R. V. Caro, Universidad de Sevilla, Sevilla, 2003, 139-146 · Zbl 1044.41020 |
| [14] | A. L. Brown, “Suns in normed linear spaces which are finite-dimensional”, Math. Ann., 279 (1987), 87-101 · Zbl 0607.41027 · doi:10.1007/BF01456192 |
| [15] | A. L. Brown, “On the connectedness properties of suns in finite dimensional spaces”, Proc. Cent. Math. Anal. Aust. Natl. Univ., 20 (1988), 1-15 · Zbl 0682.41040 |
| [16] | A. L. Brown, “On the problem of characterising suns in finite dimensional spaces”, Rend. Circ. Math. Palermo Ser. II, 68 (2002), 315-328 · Zbl 1017.41021 |
| [17] | A. S. Granero, J. P. Moreno, R. R. Phelps, “Mazur sets in normed spaces”, Discrete Comput Geom., 31 (2004), 411-420 · Zbl 1067.46018 · doi:10.1007/s00454-003-0808-5 |
| [18] | C. Franchetti, S. Roversi, Suns, M-connected sets and P-acyclic sets in Banach spaces, Preprint № 50139, Instituto di Matematica Applicata “G. Sansone”, 1988, 29 pp. |
| [19] | L. Hetzelt, “On suns and cosuns in finite-dimensional normed real vector spaces”, Acta Math. Hungar., 45:1-2 (1985), 53-68 · Zbl 0592.41043 · doi:10.1007/BF01955023 |
| [20] | V. A. Koshcheev, “The connectivity and approximative properties of sets in linear normed spaces”, Math. Notes, 17:2 (1975), 114-119 · Zbl 0327.41029 · doi:10.1007/BF01161866 |
| [21] | Å. Lima, “Banach spaces with the 4.3 intersection property”, Proc. Amer. Math. Soc., 80 (1980), 431-434 · Zbl 0452.46008 |
| [22] | J. P. Moreno, R. Schneider, “Intersection properties of polyhedral norms”, Adv. Geom., 7:3 (2007), 391-402 · Zbl 1130.52003 · doi:10.1515/ADVGEOM.2007.025 |
| [23] | I. G. Tsar’kov, “Nonunique solvability of certain differential equations and their connection with geometric approximation theory”, Math. Notes, 75:2 (2004), 287-301 · Zbl 1110.47313 |
| [24] | Sbornik: Mathematics · Zbl 0375.00007 · doi:10.4213/sm8481 |
| [25] | L. P. Vlasov, “Approximative properties of sets in normed linear spaces”, Russian Math. Surveys, 28:6 (1973), 3-66 · Zbl 0291.41028 · doi:10.1070/RM1973v028n06ABEH001624 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
