Lattice-free simplices with lattice width $2d - o(d)$
| Autor: | Mayrhofer, Lukas, Schade, Jamico, Weltge, Stefan |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: minor changes; to appear at IPCO 2022 |
| Databáze: | arXiv |
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