Universal convex coverings

Details Universal convex coverings Universal convex coverings

2008-12-18

arxiv.org/abs/0812.3525v3

Bulletin of the London Mathematical Society / The Bulletin of the London Mathematical Society 41, 6 (2010) 987-992

Mathematics

Number Theory

8 pages

Scientific paper

In every dimension $d\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set $\mathcal C\subset \mathbb R^d$ of volume at least $v_d$ and $\mathcal U_d$ contains at most $\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$.

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