Abstract
The Flatness theorem states that the maximum lattice width \(\mathrm {Flt}(d)\) 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 \(\mathrm {Flt}(d)\). 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 \(\mathrm {Flt}(d) \ge 1.138d\) arises from direct sums of a 3-dimensional lattice-free simplex.
In this work, we establish the lower bound \(\mathrm {Flt}(d) \ge 2d - O(\sqrt{d})\), 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.
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Acknowledgements
The authors would like to thank Gennadiy Averkov and Paco Santos for valuable feedback and discussions on earlier stages of this work, and Amitabh Basu for discussions about applications of the Flatness theorem within optimization. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number 451026932.
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Mayrhofer, L., Schade, J., Weltge, S. (2022). Lattice-Free Simplices with Lattice Width \(2d - o(d)\). In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_28
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