Lattice-free simplices with lattice width $2d - o(d)$

TL;DR

This paper improves the known lower bounds for the maximum lattice width of lattice-free convex sets in fixed dimensions, introducing a new family of simplices that achieve nearly linear growth with dimension.

Contribution

It establishes a new lower bound for lattice width using a novel construction based on a differential equation, and identifies local maximizers in low dimensions.

Findings

Lower bound ${ m Flt}(d) extgreater= 2d - O(\sqrt{d})$ for lattice width.

Construction of lattice-free simplices achieving this bound.

First local maximizers for 4- and 5-dimensional cases.

Abstract

The Flatness theorem states that the maximum lattice width ${\rm Flt}(d)$ 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 ${\rm Flt}(d)$. 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 ${\rm Flt}(d) \ge 1.138d$ arises from direct sums of a $3$-dimensional lattice-free simplex. In this work, we establish the lower bound ${\rm Flt}(d) \ge 2d - O(\sqrt{d})$, 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.