Small kissing polytopes

TL;DR

This paper investigates the minimal distance between disjoint lattice polytopes within a bounded coordinate space, providing explicit formulas and computational methods for various dimensions and bounds.

Contribution

It introduces an algebraic model for the lattice polytope distance and derives explicit formulas, enabling computation of previously intractable cases.

Findings

Derived explicit formula for ,k

Computed ,k for 4  8

Calculated ,k for 2  3 and ,1

Abstract

A lattice $(d,k)$-polytope is the convex hull of a set of points in $\mathbb{R}^d$ whose coordinates are integers ranging between $0$ and $k$. We consider the smallest possible distance $\varepsilon(d,k)$ between two disjoint lattice $(d,k)$-polytopes. We propose an algebraic model for this distance and derive from it an explicit formula for $\varepsilon(2,k)$. Our model also allows for the computation of previously intractable values of $\varepsilon(d,k)$. In particular, we compute $\varepsilon(3,k)$ when $4\leq{k}\leq8$, $\varepsilon(4,k)$ when $2\leq{k}\leq3$, and $\varepsilon(6,1)$.