Geodesic
A Note on Covering by Convex Bodies
Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 361-365

Voir la notice de l'article provenant de la source Cambridge University Press

A classical theorem of Rogers states that for any convex body $K$ in $n$ -dimensional Euclidean space there exists a covering of the space by translates of $K$ with density not exceeding $n\,\log \,n\,+\,n\,\log \,\log \,n\,+\,5$ . Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of $n$ the same bound can be attained by a covering which is the union of $O\left( \log \,n \right)$ translates of a lattice arrangement of $K$ .
DOI : 10.4153/CMB-2009-039-x
Mots-clés : 52C07, 52C17, 11H31 
Tóth, Gábor Fejes. A Note on Covering by Convex Bodies. Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 361-365. doi: 10.4153/CMB-2009-039-x
@article{10_4153_CMB_2009_039_x,
     author = {T\'oth, G\'abor Fejes},
     title = {A {Note} on {Covering} by {Convex} {Bodies}},
     journal = {Canadian mathematical bulletin},
     pages = {361--365},
     year = {2009},
     volume = {52},
     number = {3},
     doi = {10.4153/CMB-2009-039-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-039-x/}
}
TY  - JOUR
AU  - Tóth, Gábor Fejes
TI  - A Note on Covering by Convex Bodies
JO  - Canadian mathematical bulletin
PY  - 2009
SP  - 361
EP  - 365
VL  - 52
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-039-x/
DO  - 10.4153/CMB-2009-039-x
ID  - 10_4153_CMB_2009_039_x
ER  - 
%0 Journal Article
%A Tóth, Gábor Fejes
%T A Note on Covering by Convex Bodies
%J Canadian mathematical bulletin
%D 2009
%P 361-365
%V 52
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-039-x/
%R 10.4153/CMB-2009-039-x
%F 10_4153_CMB_2009_039_x

[1] [1] Erdős, P. and Rogers, C. A., Covering space with convex bodies. Acta Arithmetica 7(1962), 281–285. Google Scholar

[2] [2] Füredi, Z. and Kang, J.-H., Covering the n-space by convex bodies and its chromatic number. Discrete Math. 308(2008), 4495–4500 Google Scholar

[3] [3] Rogers, C. A., A note on coverings. Mathematika 4(1957), 1–6. Google Scholar

[4] [4] Rogers, C. A., Lattice coverings of space: the Minkowski–Hlawka theorem. Proc. London Math. Soc. 8(1958), 447–465. Google Scholar

[5] [5] Rogers, C. A., Lattice coverings of space. Mathematika 6(1959), 33–39. Google Scholar

[6] [6] Schmidt, W., Masstheorie in der Geometrie der Zahlen. Acta Mathematica 102(1959), 159–224. Google Scholar

[7] [7] Siegel, C. L., A mean value theorem in geometry of numbers. Ann. of Math. 46(1945), 340–347. Google Scholar

Cité par Sources :