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Bulletin of the London Mathematical Society Advance Access published online on September 3, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp076
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© 2009 London Mathematical Society

Universal convex coverings

Roland Bacher

Institut Fourier
Laboratoire de Mathématiques UMR 5582 (UJF-CNRS)
BP 74
38402 St Martin d’Hères cedex
France

Received 12 January 2009.

In every dimension d ≥ 1, we establish the existence of a constant vd > 0 and of a discrete subset Ud of Rd such that the following holds: C + Ud = Rd for every convex set C sub Rd of volume at least vd and such that Ud contains at most log (r)d–1rd points at distance at most r from the origin, for every large r.

2000 Mathematics Subject Classification 11H06, 52A45, 52C17.


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