Kissing polytopes in dimension 3

TL;DR

This paper determines the exact minimal distance between disjoint lattice polytopes within a cube in three dimensions, using a novel polynomial root-finding approach based on lattice point modeling.

Contribution

It introduces a precise formula for the minimal distance between lattice polytopes in 3D and develops a computational method involving polynomial root analysis.

Findings

Exact minimal distance formula for lattice polytopes in 3D

Reduction of the problem to polynomial root calculations

Application of symbolic computation for precise results

Abstract

It is shown that the smallest possible distance between two disjoint lattice polytopes contained in the cube $[0,k]^3$ is exactly $$ \frac{1}{\sqrt{2(2k^2-4k+5)(2k^2-2k+1)}} $$ for every integer $k$ at least $4$. The proof relies on modeling this as a minimization problem over a subset of the lattice points in the hypercube $[-k,k]^9$. A precise characterization of this subset allows to reduce the problem to computing the roots of a finite number of degree at most $4$ polynomials, which is done using symbolic computation.