Universal convex coverings

Roland Bacher (1)

1 IF - Institut Fourier

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 $r$.

Mots clés: Convex set; Covering

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]; Géométrie métrique [math.MG]

Roland Bacher. Universal convex coverings. Bulletin of the London Mathematical Society, 2010, 41 (6), pp.987-992. ⟨hal-00347942v3⟩