A reverse Minkowski theorem.

We prove a conjecture due to Dadush, showing that if L⊂ ℝn is a lattice such that det(L′) ≥ 1 for all sublattices L′ ⊆ L, then where t := 10(logn + 2). This implies bounds on the number of lattice points in Euclidean balls for various different radii, which can be seen as a reverse form of Minkowski’s First Theorem.