Kissing Polytopes in Discrete Geometry

• Kissing polytopes are disjoint convex figures in Euclidean spaces or lattices that approach each other as closely as geometric and combinatorial constraints allow.
• Analytical methods employing lattice coordinates, determinant estimates, and algebraic optimization provide exact formulas and asymptotic bounds for their minimal Euclidean distances.
• These studies enhance understanding in discrete optimization and packing problems by refining the concepts of translative and lattice kissing numbers.

A kissing polytope is a configuration in Euclidean space or on a lattice in which two convex polytopes are disjoint but approach each other as closely as permitted by given combinatorial or geometric constraints. The central problem is to determine the minimal Euclidean distance that can be achieved between two disjoint lattice polytopes contained in a fixed hypercube, or more generally to enumerate, bound, and construct maximal configurations of polytopes that “kiss” a central copy or each other. This notion generalizes the classical kissing-number problem for spheres and plays a foundational role in discrete geometry, extremal combinatorics, and optimization theory.

1. Foundational Definitions and Models

In lattice polytopes, a $(d,k)$-polytope is the convex hull of points in $\mathbb{R}^d$ with integer coordinates in $[0, k]$ (Deza et al., 2024). The kissing distance $\varepsilon(d,k)$ is the smallest possible Euclidean distance between two disjoint lattice $(d,k)$-polytopes: $$\varepsilon(d,k) = \min \left\{ \min_{p \in P,\, q \in Q} \|p-q\| : P, Q \subset [0,k]^d \text{ are disjoint lattice polytopes} \right\}$$ This definition extends to rational polytopes, with additional complexity parameters controlling the bounds (Deza et al., 2023).

In the context of translative and lattice kissing numbers for polytopes such as cross-polytopes, one considers the size of maximal configurations where translated or lattice copies touch a central polytope along their boundary but remain otherwise disjoint. For the $n$-dimensional cross-polytope $$K_n = \{x \in \mathbb{R}^n : \sum_{i=1}^n |x_i| \le 1\}$$, the translative kissing number $\kappa_T(K_n)$ is the size of the largest set $X\subset \mathbb{R}^n$ such that the interiors of translates $x+\frac12 K_n$ are pairwise disjoint and each boundary touches the central copy (Miller, 16 Jan 2025).

2. Exact Formulas and Asymptotic Bounds

Results for kissing polytopes frequently involve sharp lower and upper bounds and, in selected cases, closed-form expressions for minimal distance.

• Dimension 2: The exact formula is

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

  for all $k\ge2$ (Deza et al., 2024).

• Dimension 3: For $k \ge 4$, the minimal kissing distance is

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

  (Deza et al., 26 Feb 2025). For point-to-triangle separation, a sharper bound is

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

  for $k\ge2$ (Deza et al., 6 Jan 2026).

• Higher Dimensions and General Lattice Bounds: For $d \ge 4$ and $k \ge 2$,

  $$\varepsilon^u(d,k) \ge \frac{1}{k^{d-1}(\sqrt{d})^{d}}$$

  reflecting exponential decay in $d$ at fixed $k$ (Deza et al., 6 Jan 2026).

• Cross-polytope Kissing Number: Upper bound

  $\kappa_T(K_n) \le 2.9162^{(1+o(1))n}$

  and lower bound

  $\kappa_T(K_n) \ge 1.1637^{(1-o(1))n}$

  for large $n$, strictly improving classical bounds such as Hadwiger’s $3^n-1$ (Miller, 16 Jan 2025).

The following table presents selected exact values:

Dimension $d$	$k$	Minimal Distance $\varepsilon(d,k)$	Reference
2	Any $\ge2$	$\frac{1}{\sqrt{(k-1)^2 + k^2}}$	(Deza et al., 2024)
3	$\ge4$	$\frac{1}{\sqrt{2(2k^2-4k+5)(2k^2-2k+1)}$	(Deza et al., 26 Feb 2025)
3 (pt-tri)	$\ge2$	$\frac{1}{\sqrt{3k^4 - 4k^3 + 4k^2 - 2k + 1}}$	(Deza et al., 6 Jan 2026)
4	2	$\frac{1}{\sqrt{209}}$	(Deza et al., 6 Jan 2026)
4	Any	$\varepsilon^u(4,k) \ge 1/(k^3 \cdot 8)$	(Deza et al., 6 Jan 2026)

3. Algebraic and Geometric Methodologies

The solution of kissing polytopes problems is grounded in discrete optimization, matrix analysis, and combinatorial geometry. The minimal kissing distance can typically be reformulated as an integer optimization over affine hulls of lattice simplices. For $d$-dimensional polytopes,

• Reduction to pairs of opposite faces of a lattice simplex realizes the extremal distance (Deza et al., 6 Jan 2026, Deza et al., 2024).
• For dimension 3, the algebraic model uses a $3\times 2$ integer matrix $A$ and offset $b$ to encode the pair $(P,Q)$, reducing the problem to minimal value analysis of

  $$d(P,Q) = |\text{cubic form}| / \sqrt{\text{quartic form}}$$

  over finite lattice patterns (Deza et al., 26 Feb 2025).

• The symbolic computation and pattern enumeration (enabled by cube symmetries) restrict candidate extremal configurations to a manageable, finite set (Deza et al., 26 Feb 2025, Deza et al., 2024).

Similar constructions yield bounds in higher dimensions via determinant estimates (Hadamard’s inequality), convex geometry, and entropy function optimization (Miller, 16 Jan 2025, Deza et al., 2023).

4. Kissing Numbers for Standard Polytopes and Sphere Packings

Kissing polytope problems generalize classical kissing-number questions and provide combinatorial insight into extremal packing arrangements.

• Classical Kissing Numbers: Known for spheres in dimensions $n=1,2,3,4,8,24$ with values $2,6,12,24,240,196560$, respectively (Altschuler et al., 2013).
• Regular Convex Polytopes: In dimension 3, the regular icosahedron provides the unique maximal (12-point) kissing polytope. In 4D, the unique 24-cell underlies all 24-point sphere kissing configurations (Altschuler et al., 2013).
• Dimension 5: The explicit construction of a 40-point arrangement $Q_5$ in $\mathbb{R}^5$ demonstrates the existence of new non-isometric maximal kissing polytopes whose combinatorics and symmetry differ sharply from lower-dimensional analogues (Szöllősi, 2023).

Kissing configurations and their convex hulls encode information about admissible separation, antipodality, and symmetry properties, central to the study of lattice packings and extremal discrete sets.

5. Lattice and Translative Kissing Configurations

The distinction between translative and lattice kissing numbers in cross-polytopes yields refined asymptotic and uniqueness results.

• Translative Kissing Number $\kappa_T(K_n)$: Maximizes the size of a set of translates mutually kissing a central copy. Miller’s work achieves bounds of $2.9162^{(1+o(1))n}$ (upper) and $1.1637^{(1-o(1))n}$ (lower), surpassing prior results (Miller, 16 Jan 2025).
• Lattice Kissing Number $\kappa_L(K_n)$: Counts maximal lattice vectors of minimal norm lying on the polytope boundary. For $n\ge1$, $\kappa_L(K_n) < 12(2^n-1)$ (Miller, 16 Jan 2025).
• Uniqueness in 4D: The unique lattice achieving the maximal value $\kappa_L(K_4)=40$ is the even unimodular half-shift lattice $D_4^+$, up to signed permutations. The proof involves support-size arguments and covering-radius lemmas (Miller, 16 Jan 2025).
• These results place the cross-polytope problem alongside sphere kissing numbers in quantitative sharpness and methodological similarity.

6. Computational Complexity, Applications, and Extensions

The minimal kissing distance between disjoint polytopes governs complexity bounds for optimization algorithms and quantitative geometry.

• Exact Computation: Algorithms reduce to enumeration of all simplex pairs with dimensions summing to $d-1$, followed by quadratic minimization. For fixed $d$, these are polynomial in $(n+1)^d$ (Deza et al., 2023).
• Lower and Upper Bounds: Determinant-based lower bounds (using Hadamard’s inequality) yield exponential decay in $d$:

  $\delta(P,Q) \ge \frac{1}{(n d)^{2d}}$

  Matching upper bounds constructed by explicit block-constant polytopes confirm rate optimality (Deza et al., 2023).

• Bounding Complexity Constants: In polytopal optimization, complexity constants (facial distance, number of oracle calls) degrade as inverse powers of the kissing distance. For the conditional-gradient (Frank-Wolfe) method, the minimal separation dictates convergence rates, and for polytope intersection certification, required calls scale as $O(\delta^{-2})$ (Deza et al., 2023).

7. Open Questions and Further Directions

While closed-form solutions are available for low dimensions, identifying infinite families of extremal kissing polytopes in arbitrary $(d, k)$ remains unresolved.

• High-dimensional asymptotics: A plausible implication is that $\varepsilon(d,k) = \Theta(k^{-(d-1)})$ as $k\to\infty$ (Deza et al., 2024, Deza et al., 6 Jan 2026).
• Uniqueness and classification: In dimensions 4 and 5, uniqueness of maximal kissing polytopes is only partially established (Altschuler et al., 2013, Szöllősi, 2023). The combinatorial and geometric structures of new configurations such as $Q_5$ warrant further study.
• Symbolic computation and enumeration: Advances in algebraic modeling and finite pattern reduction have enabled computation in dimensions $d=3,4,6$ for selected $k$, but closed forms for all $k$ are rare (Deza et al., 2024, Deza et al., 26 Feb 2025).
• Role in discrete optimization: The minimal distance bounds have direct impact in the worst-case analysis of geometric algorithms and quantification of extremal combinatorial parameters.

Kissing polytopes thus represent a central paradigm at the interface of discrete geometry, extremal optimization, and asymptotic combinatorics, with sharp results, explicit constructions, and remaining conjectures guiding ongoing research.