Improved bounds on the diameter of lattice polytopes

TL;DR

This paper establishes improved upper bounds on the maximum diameter of lattice polytopes with integer vertices within a bounded range, confirming a conjecture and providing exact values in specific cases.

Contribution

It proves a new upper bound on the diameter of lattice polytopes for k ≥ 3 and confirms the conjectured maximum for 4-dimensional polytopes with k=3.

Findings

Maximum diameter bound: kd - ⌈2d/3⌉ for k ≥ 3

Exact diameter: δ(4,3) = 8

Supports the conjecture δ(d,k) ≤ ⌊(k+1)d/2⌋

Abstract

We show that the largest possible diameter $\delta(d,k)$ of a $d$-dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rceil$ when $k\geq3$. In addition, we show that $\delta(4,3)=8$. This substantiates the conjecture whereby $\delta(d,k)$ is at most $\lfloor(k+1)d/2\rfloor$ and is achieved by a Minkowski sum of lattice vectors.