Kissing Polytopes Distance in Lattice Geometry

• Distance of kissing polytopes is the minimal Euclidean separation between disjoint lattice polytopes confined to a hypercube, a critical metric in discrete geometry.
• The study uses algebraic models and reductions to lattice simplices to derive exact formulas and tight asymptotic bounds in low dimensions.
• Computational enumeration and determinant-based lower bounds inform practical strategies for optimizing separation guarantees in high-dimensional polyhedral analysis.

The distance of kissing polytopes quantifies the minimal separation between two disjoint lattice polytopes contained in a bounded hypercube, capturing a fundamental extremal metric in discrete geometry with significant ramifications for polyhedral combinatorics and algorithmic complexity. This problem is formalized as the determination of the smallest possible Euclidean distance—termed the kissing distance—between any two disjoint polytopes whose vertices are lattice points inside $[0,k]^d$ for fixed integers $d$ (dimension) and $k$ (hypercube size). This distance has been analyzed through algebraic models, explicit constructions, determinant-based lower bounds, and computational enumeration, with comprehensive exact results in low dimensions and new advances in tight asymptotic and explicit formulae.

1. Fundamental Definitions and Problem Statement

Let $[0,k]^d \subset \mathbb{R}^d$ denote the $d$-dimensional axis-aligned hypercube of edge length $k$. A lattice $(d,k)$-polytope is a convex polytope $P \subset [0,k]^d$ with all vertices in $\mathbb{Z}^d$. The kissing distance $\varepsilon(d,k)$ is the minimal Euclidean distance between any two disjoint lattice $(d,k)$-polytopes $P$ and $Q$: $$\varepsilon(d,k) = \min \{ \|p - q\| : p \in P,\, q \in Q,\, P \cap Q = \emptyset \}$$ A pair attaining this minimum is a kissing pair. The central question is to determine or tightly bound $\varepsilon(d,k)$ as a function of $d$ and $k$. This extremal value is critical in settings such as optimization, where it controls algorithmic separation guarantees, especially in high-dimensional or high-complexity regimes (Deza et al., 2023).

2. Algebraic Models and Reduction to Simplices

The computation of $\varepsilon(d,k)$ admits a significant simplification via reduction to pairs of disjoint lattice simplices with complementary dimensions: there always exists a kissing pair of simplices $P, Q \subset [0,k]^d$ with $\dim P + \dim Q = d-1$ for which $\varepsilon(d,k)$ is achieved (Deza et al., 2024, Deza et al., 2023, Deza et al., 6 Jan 2026). Given such $P$ (vertices $p^0, ..., p^n$) and $Q$ (vertices $q^0, ..., q^m$), with $n+m = d-1$, their squared Euclidean distance can be written as the minimum of a quadratic form: $$d(P,Q)^2 = \min_{\lambda \in \Delta_n,\, \mu \in \Delta_m} \left\| \sum_{i=0}^{n} \lambda_i p^i - \sum_{j=0}^{m} \mu_j q^j \right\|^2$$ where $(\lambda_i)$, $(\mu_j)$ are barycentric coordinates in the respective simplices. This yields the least-squares matrix expression

$\|A(A^T A)^{-1}A^T b - b\|^2$

where $A$ is the $d \times (d-1)$ difference matrix and $b$ the offset vector encoding the affine displacement between $P$ and $Q$ (Deza et al., 2024).

3. Exact Formulas and Special-Dimensional Results

The determination of $\varepsilon(d,k)$ is fully explicit for small $d$, with sharp algebraic expressions and canonical extremal constructions.

• Dimension 2: The minimal distance is between a lattice point and a disjoint lattice segment. The closed formula is:

$\varepsilon(2,k) = \frac{1}{\sqrt{(k-1)^2 + k^2}}$

attained (up to symmetry) by the point $(1,1)$ and the segment joining $(0,0)$ to $(k,k-1)$ (Deza et al., 2024).

• Dimension 3: For all $k \ge 4$, the unique minimizing pair consists of two lattice segments, explicitly given by $P^* = \mathrm{conv}\{ (k,2,1),\, (0,k-1,k) \}$ and $Q^* = \mathrm{conv}\{ (0,0,0),\, (k-1,k,k) \}$. The formula is:

$$\varepsilon(3,k) = \frac{1}{\sqrt{2(2k^2-4k+5)(2k^2-2k+1)}}$$

This minimal value arises from a constrained combinatorial optimization over the discrete parameter space and is certified as uniquely extremal (up to cube symmetries) by algebraic and computational arguments (Deza et al., 26 Feb 2025). For point-to-triangle distances, a separate explicit formula is attained by the configuration $P = (1,1,1)$ and $$Q = \mathrm{conv}\{ (0,0,1),\, (k,k-1,0),\, (0,k,k) \}$$:

$$\varepsilon_0(3,k) = \frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}}$$

but this is not minimal among general polytope pairs (Deza et al., 6 Jan 2026).

The minimal distance in increasing $k$ exhibits polynomial decay, with $k^{-2}$ asymptotics evident in three dimensions.

4. Determinant-Based Lower Bounds and Asymptotics

Optimal kissing distances in higher dimensions are controlled by determinants of Gram matrices arising from the configuration matrices $A$, leading to general lower bounds: $\varepsilon(d,k) \ge \frac{1}{k^{d-1} d^{d/2}}$ This bound is obtained by applying Hadamard's inequality to the minors of $A$ and refining previous work that yielded a $d^{(d+1)/2}$ scaling in the denominator (Deza et al., 6 Jan 2026). In the special case $k=1$ (unit cube), this approach recovers the Alon–Vu / Graham–Sloane style exponential bound for simplex distances: $\varepsilon_0(d,1) \ge \frac{2^{d-1}}{d^{(d+1)/2}}$ This suggests significantly improved dependence on $d$ in new estimates. The tightness and sharpness of these determinant-based lower bounds in large $d$ or $k$ is an open aspect, as explicit constructions are lacking for $d \ge 4$.

5. Computational Methods and Algorithmic Strategies

The brute-force enumeration of kissing pairs is prohibitive due to the exponential number of simplices in $d, k$. Efficient computational strategies exploit algebraic reduction, symmetry, and number-theoretic normalization. The row-wise enumeration methodology, introduced by Deza–Liu–Pournin (Deza et al., 2024), generates all possible difference vectors (rows) modulo greatest common divisor and sign, then assembles candidate matrices $(A,b)$ by selecting $d$ distinct rows. The invertibility of $A^T A$ is tested numerically, and the value of $\|A(A^T A)^{-1}A^T b - b\|$ is computed. No explicit symmetry reduction is necessary, as the process naturally eliminates redundant configurations.

For moderate $d$ and small $k$, this reduces computational burden by several orders of magnitude. In three dimensions, the final reduction to quartic polynomial comparison on a finite set of candidate configurations enables complete symbolic verification and exact calculation (Deza et al., 26 Feb 2025).

6. Theoretical Insights and Open Problems

Key structural theorems establish that the minimal distance is always realized by pairs of simplices with complementary dimensions. The index $i$ of the minimizing opposite face in higher dimensions remains uncharacterized in general; only special cases (e.g., $i=0$ for $d=2$, $i=1$ for $d=3$) are resolved (Deza et al., 6 Jan 2026). Monotonicity results confirm that $\varepsilon(3,k) < \varepsilon(2,k)$ for all $k$, and that segment-segment configurations dominate over point-triangle pairs in three dimensions (Deza et al., 26 Feb 2025).

Open questions include:

• Precise determination of $\varepsilon(d,k)$ for $d \ge 4$.
• Structural characterization of all kissing pairs in high dimension.
• Asymptotic expansion of $\varepsilon(d,k)$ as $k \to \infty$ for fixed $d$.
• Extension of results to lattices other than $\mathbb{Z}^d$ or to non-standard hypercube domains.
• Advancement of determinant-based techniques, possibly via Minkowski-type arguments, to further sharpen lower bounds and approach the true asymptotics.

A plausible implication is that continued refinement in the algebraic and combinatorial modeling of the problem, as well as more efficient enumeration algorithms, may yield new exact values or tighten asymptotic analysis in higher dimensional and parameter regimes.

7. Impact and Applications

The distance of kissing polytopes plays a foundational role in the complexity analysis of convex optimization algorithms, particularly those that require quantitative separation bounds for algorithmic intersection tests. In primal-dual and cutting-plane frameworks, the complexity of certifying disjointness scales with $1/\varepsilon(d,k)^2$, so doubly exponential decay with $d$ or encoding length $L$ implies the necessity of exponential oracle calls in the worst case (Deza et al., 2023). These results inform both theoretical limitations and practical expectations for optimization over high-dimensional polyhedral feasible regions.

Furthermore, the geometric and combinatorial properties of kissing polytopes interface with extremal questions in lattice theory, polyhedral combinatorics, and discrete convexity, situating this metric as a focal point in the structure theory of lattice polytopes and their algorithmic applications.