Lattice-Free Simplices with Lattice Width $$2d - o(d)$$

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