# Computational determination of the largest lattice polytope diameter

## TL;DR

This paper develops a computational approach to determine the maximum diameter of lattice polytopes in small dimensions, confirming conjectures for specific cases in three dimensions.

## Contribution

It introduces a computational framework to find the largest lattice polytope diameter and verifies conjectured bounds for (3,4) and (3,5) cases.

## Key findings

- (3,4) = 7
- (3,5) = 9
- Confirmed the conjecture relating diameter to (k+1)d/2

## Abstract

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. We show that {\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors.