Geodesic
Incoherent systems and coverings in finite dimensional Banach spaces
Sbornik. Mathematics, Tome 205 (2014) no. 5, pp. 703-721 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We discuss the construction of coverings of the unit ball of a finite dimensional Banach space. There is a well-known technique based on comparing volumes which gives upper and lower bounds on covering numbers. However, this technique does not provide a method for constructing good coverings. Here we study incoherent systems and apply them to construct good coverings. We use the following strategy. First, we build a good covering using balls with a radius close to one. Second, we iterate this construction to obtain a good covering for any radius. We shall concentrate mainly on the first step of this strategy. Bibliography: 14 titles.
Keywords: incoherent systems, covering of balls, Banach space, modulus of smoothness
Mots-clés : explicit constructions. 
@article{SM_2014_205_5_a5,
     author = {V. N. Temlyakov},
     title = {Incoherent systems and coverings in finite dimensional {Banach} spaces},
     journal = {Sbornik. Mathematics},
     pages = {703--721},
     year = {2014},
     volume = {205},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2014_205_5_a5/}
}
TY  - JOUR
AU  - V. N. Temlyakov
TI  - Incoherent systems and coverings in finite dimensional Banach spaces
JO  - Sbornik. Mathematics
PY  - 2014
SP  - 703
EP  - 721
VL  - 205
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_2014_205_5_a5/
LA  - en
ID  - SM_2014_205_5_a5
ER  - 
%0 Journal Article
%A V. N. Temlyakov
%T Incoherent systems and coverings in finite dimensional Banach spaces
%J Sbornik. Mathematics
%D 2014
%P 703-721
%V 205
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2014_205_5_a5/
%G en
%F SM_2014_205_5_a5
V. N. Temlyakov. Incoherent systems and coverings in finite dimensional Banach spaces. Sbornik. Mathematics, Tome 205 (2014) no. 5, pp. 703-721. http://geodesic.mathdoc.fr/item/SM_2014_205_5_a5/

[1] K. Borsuk, “Drei Sätze über die $n$-dimensionale euclidische Sphäre”, Fund. Math., 20 (1933), 177–190 | Zbl

[2] J. Bourgain, J. Lindenstrauss, “On covering a set in $\mathbb R^N$ by balls of the same diameter”, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., Springer, Berlin, 1991, 138–144 | DOI | MR | Zbl

[3] J. Kahn, G. Kalai, “A counterexample to Borsuk's conjecture”, Bull. Amer. Math. Soc. (N.S.), 29:1 (1993), 60–72 | DOI | MR | Zbl

[4] N. Alon, “Discrete mathematics: methods and challenges”, Proceedings of the International Congress of Mathematicians (Beijing, 2002), v. I, Higher Ed. Press, Beijing, 2002, 119–135 | MR | Zbl

[5] A. Bondarenko, On Borsuk's conjecture for two-distance sets, 2013, 9 pp., arXiv: 1305.2584v3

[6] A. M. Raigorodskii, Gipoteza Knezera i topologicheskii metod v kombinatorike, MTsNMO, M., 2011, 28 pp.

[7] V. N. Temlyakov, “Greedy approximations”, Foundations of computational mathematics (Santander 2005), London Math. Soc. Lecture Note Ser., 331, Cambridge Univ. Press, Cambridge, 2006, 371–394 | DOI | MR | Zbl

[8] V. N. Temlyakov, Greedy approximation, Cambridge Monogr. Appl. Comput. Math., 20, Cambridge Univ. Press, Cambridge, 2011, xiv+418 pp. | DOI | MR | Zbl

[9] G. G. Lorentz, M. v. Golitschek, Yu. Makovoz, Constructive approximation. Advanced problems, Grundlehren Math. Wiss., 304, Springer-Verlag, Berlin, 1996, xii+649 pp. | MR | Zbl

[10] J. H. van Lint, R. M. Wilson, A course in combinatorics, 2nd ed., Cambridge Univ. Press, Cambridge, 2001, xiv+602 pp. | MR | Zbl

[11] Xiteng Liu, Sparse signal representation in redundant systems, Ph. D. Dissertation, University of South Carolina, 2006, 75 pp. | MR

[12] C. A. Rogers, “Covering a sphere with spheres”, Mathematika, 10:2 (1963), 157–164 | DOI | MR | Zbl

[13] J.-L. Verger-Gaugry, “Covering a ball with smaller equal balls in $\mathbb R^n$”, Discrete Comput. Geom., 33:1 (2005), 143–155 | DOI | MR | Zbl

[14] I. Dumer, “Covering spheres with spheres”, Discrete Comput. Geom., 38:4 (2007), 665–679 | DOI | MR | Zbl