@inproceedings{0c24182239de4fc781623c676f916e1a,
title = "Lattice-Free Simplices with Lattice Width 2 d- o(d)",
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{\textquoteright}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) ≥ 1.138 d 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.",
keywords = "Flatness theorem, Lattice-free, Simplices",
author = "Lukas Mayrhofer and Jamico Schade and Stefan Weltge",
note = "Publisher Copyright: {\textcopyright} 2022, Springer Nature Switzerland AG.; 23rd International Conference on Integer Programming and Combinatorial Optimization, IPCO 2022 ; Conference date: 27-06-2022 Through 29-06-2022",
year = "2022",
doi = "10.1007/978-3-031-06901-7_28",
language = "English",
isbn = "9783031069000",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Science and Business Media Deutschland GmbH",
pages = "375--386",
editor = "Karen Aardal and Laura Sanit{\`a}",
booktitle = "Integer Programming and Combinatorial Optimization - 23rd International Conference, IPCO 2022, Proceedings",
}