Small kissing polytopes
Antoine Deza, Zhongyuan Liu, Lionel Pournin
TL;DR
This work investigates the minimal distance $\varepsilon(d,k)$ between disjoint lattice $(d,k)$-polytopes, linking it to optimization concepts and providing an algebraic, least-squares framework to bound and compute it. The authors derive a quadratic objective $f_{P,Q}(\lambda,\mu)$ via a matrix $A$ and vector $b$ built from polytope vertices, show when the minimum is unique, and use a relaxation to obtain a lower bound that equals the affine-hull distance under certain coefficient conditions. They obtain a closed-form expression for the two-dimensional case: $\varepsilon(2,k)=\frac{1}{\sqrt{(k-1)^2+k^2}}$ for $k>1$, and demonstrate sharpness with explicit kissing polytopes. Using a coordinate-wise enumeration strategy that builds a row list $\mathcal{L}$ and selects $d$ rows to form $A,b$, they compute previously intractable values: $\varepsilon(3,k)$ for $4\le k\le 8$, $\varepsilon(4,k)$ for $2\le k\le 3$, and $\varepsilon(6,1)$, each paired with explicit kissing polytopes, and they report that all computed bounds are exact. This advances the practical computation of $\varepsilon(d,k)$ and enriches the catalogue of kissing lattice polytopes with concrete constructions.
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 $\varepsilon(6,1)$.