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

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DOI

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
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     title = {A {Note} on {Covering} by {Convex} {Bodies}},
     journal = {Canadian mathematical bulletin},
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     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/}
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