No abstract

The Flatness theorem states that the maximum lattice width Flt(d) of a d-dimensional lattice-free convex set is finite. It is the key ingredient for Lenstra's algorithm for integer programming in fixed dimension, and much work has been done to obtain bounds on Flt(d). While most results have been concerned with upper bounds, only few techniques are known to obtain lower bounds. In fact, the previously best known lower bound Flt(d) ≥ 1.138d arises from direct sums of a 3-dimensional lattice-free simplex. In this work, we establish the lower bound Flt(d) ≥ 2d − O( √ d), attained by a family of lattice-free simplices. Our construction is based on a differential equation that naturally appears in this context. Additionally, we provide the first local maximizers of the lattice width of 4-and 5-dimensional lattice-free convex bodies.