Kissing polytopes in dimension 3
Antoine Deza, Zhongyuan Liu, Lionel Pournin
TL;DR
This paper determines the exact minimal distance between disjoint lattice $(3,k)$-polytopes inside the cube for all $k\neq 3$, proving the distance is $\varepsilon(3,k)=1/\sqrt{2(2k^2-4k+5)(2k^2-2k+1)}$. The authors recast the geometric problem as a discrete optimization over lattice points in $[-k,k]^9$, expressing the distance bound as $\frac{|f(x)|}{\sqrt{g(x)}}$ where $g(x)=(x_1x_5-x_2x_4)^2+(x_1x_6-x_3x_4)^2+(x_2x_6-x_3x_5)^2$ and $f(x)=x_1(x_6x_8-x_5x_9)+x_2(x_4x_9-x_6x_7)+x_3(x_5x_7-x_4x_8)$. The analysis shows point–triangle configurations cannot achieve the minimum for $k\geq8$, reducing the problem to pairs of line segments. For $k\geq6$ the search further reduces to a finite, $k$-independent set, and symbolic computation identifies eight representative lattice points corresponding to the extremal pair $P^*,Q^*$; this yields the closed-form distance and confirms the formula across all admissible $k$, with small-$k$ cases verified separately. The results provide exact kissing-distance values in 3D, offering precise stopping criteria for projection-based algorithms and contributing to the combinatorial geometry of lattice polytopes.
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.