Packing Minima in Geometry & Optimization

• Packing minima are measures that quantify the minimal scaling required for non-overlapping arrangements in convex bodies and lattice structures.
• They arise in diverse applications including discrete geometry, computational packing, and optimization, establishing sharp volume bounds and lattice point inequalities.
• Algorithms leveraging packing minima address NP-hard container problems and guide theoretical insights in learning theory and high-dimensional function spaces.

Packing minima formalize extremal questions in geometry, optimization, and combinatorics where one seeks the most compact non-overlapping arrangement of a prescribed family of objects under given constraints. The concept arises in multiple mathematical theories, including discrete geometry, the geometry of numbers, computational packing, and optimization in both continuous and discrete settings. Packing minima typically quantify minimal radii, minimal container sizes, minimal densities, or minimal parameter values that enforce non-overlap, often indexed by combinatorial, algebraic, or geometric invariants.

1. Formal Definition and Context in Geometry of Numbers

Packing minima, denoted $\rho_i(K, \Lambda)$ for $1 \leq i \leq n$, associate to a convex body %%%%2%%%% containing the origin and a full-rank lattice $\Lambda\subset\mathbb{R}^n$. The $i$-th packing minimum is given by

$$\rho_i(K, \Lambda) := \inf \left\{ \rho>0 \; | \; (\rho K + L) \cap \Lambda \neq L \cap \Lambda \text{ for every } (n-i)\textrm{-plane } L \right\}$$

or equivalently,

$$\rho_i(K, \Lambda) = \max_{L} \lambda_1(K|L^\perp, \Lambda|L^\perp)$$

where $L$ runs over all $(n-i)$-dimensional subspaces of $\mathbb{R}^n$, $K|L^\perp$ denotes the orthogonal projection, and $\lambda_1$ is the first successive minimum (Henk et al., 2020, Han et al., 20 Jan 2026). These minima quantify the minimal scaling factors along directions in which new lattice points may appear, directly controlling the geometry and density of packings.

Packing minima serve as duals to covering minima (Kannan–Lovász), interpolating between the successive minima of $K$ and the reciprocals of those of the polar body $K^*$. In particular, when $K$ is origin-symmetric:

$$\frac{1}{\lambda_i(K^*, \Lambda^*)} \leq \rho_i(K, \Lambda) \leq \lambda_{n-i+1}(K, \Lambda)$$

which refines Minkowski-type inequalities and transference theorems (Henk et al., 2020).

2. Volume and Lattice Point Inequalities for Packing Minima

Sharp volume bounds analogous to Minkowski’s second theorem have been established for packing minima. If $K$ is centered and $\Lambda$ a rank-$n$ lattice, then

$$\frac{(n+1)}{n!}\prod_{i=1}^n \frac{1}{\rho_i(K, \Lambda)} \leq \frac{\operatorname{vol}_n(K)}{\det \Lambda} \leq \prod_{i=1}^n \frac{1}{\rho_i(K, \Lambda)}$$

with equality for generalized cross-polytopes (specific axis-aligned boxes) (Han et al., 20 Jan 2026).

Discrete analogues relate packing minima to the count of lattice points:

$$|K \cap \Lambda| \leq \prod_{i=1}^n \left\lfloor \frac{1}{\rho_i(K, \Lambda)} + 1 \right\rfloor$$

and, in two dimensions,

$$|K \cap \mathbb{Z}^2| \geq \frac{1}{2} \left\lfloor \frac{1}{\rho_1} - 2 \right\rfloor\left(\frac{1}{\rho_2} - 2\right) - 2$$

demonstrating near-tightness for standard bodies and explicit cases (Henk et al., 2020).

These inequalities are sharp for cubes, cross-polytopes, and centered simplices, where packing minima are explicitly computable (Han et al., 20 Jan 2026, Henk et al., 2020). For instance, the centered simplex $T_n$ yields $$\rho_i(T_n, \mathbb{Z}^n) = \frac{1}{\left\lfloor n/(i+1) \right\rfloor + 1}$$.

3. Computational Packing Minima: Algorithms and Approximability

Packing minima arise naturally in combinatorial and geometric optimization, e.g., minimal container problems, recursive shape packings, and compact bin packing. These problems are typically NP-hard, with the minimum possible container size (area, volume, etc.) acting as the packing minimum (Alt et al., 2016, Huang et al., 2014).

Approximation algorithms achieve constant-factor bounds for minimal container volumes in three dimensions for various object families and motion models (Alt et al., 2016). For axis-parallel boxes packed by translation into a box, a $(\alpha+\varepsilon)$-approximation (where $\alpha$ arises from a 2D packing subroutine) is attainable via metric rounding and strip-packing decomposition, with similar results for arbitrary convex objects and motion constraints.

$$V_{\text{alg}} \leq (\alpha+\varepsilon)V_\text{opt}$$

for optimized parameters, with explicit constants: $7.25$ for axis-parallel boxes, $11.54$ for height-bucketing, and up to $511.37$ for convex polyhedra in convex containers.

Online packing minima for area or perimeter reveal worst-case competitive ratios: perimeter minimization admits algorithms with absolute ratios $<4$, while area minimization is subject to $\Omega(\sqrt{n})$ lower bounds and $O(\sqrt{n})$ matching algorithms (Abrahamsen et al., 2021).

4. Packing Minima in Extremal Discrete Geometry and Shape Theory

Packing minima provide a language for extremal questions regarding pessimal and optimal packing densities. In shape optimization, the study of local minima in packing density—for example, global or local pessimal shapes—relies on packing minima and their directional derivatives (Kallus, 2012, Kallus, 2013).

The sphere is a local pessimum for lattice packing in three dimensions: for any origin-symmetric convex body $K$ close to the ball $B^3$, $\varphi(K) > \varphi(B^3) = \frac{\pi}{\sqrt{18}}$, as established by first-order perturbation arguments and the extremal structure of the face-centered cubic lattice (Kallus, 2012). In higher dimensions, the sphere loses this property due to eutaxy and perfection properties of critical lattices.

Further, in the context of totally separable or $\rho$-separable packings, packing minima encode the minimal mean projection (Kubota–Alexandrov–Fenchel chain), with ball-like shapes arising as near minimizers for large $n$ (Bezdek et al., 2017). The geometry of packing minima thus determines asymptotic and local extremal configurations.

5. Packing Minima in Algorithmic and Applied Settings

Continuous optimization for packing minima is evident in robotics and computational geometry, such as SDF-Pack (Pan et al., 2023), where a truncated signed-distance field is minimized to achieve tight, collision-free placements of objects in bins. Here, the container volume or packed object count become practical proxies for packing minima, and algorithms are designed to minimize such metrics via geometric heuristics.

In recursive circle-packing (RCPP), Dantzig–Wolfe reformulations exploit packing minima at each recursion level to derive strong dual bounds and globally optimal or near-optimal packings (Gleixner et al., 2017). Verified patterns (packing minima at each packing level) guarantee tight primal and dual solutions.

Sphere and circle packing minima also organize computational approaches for unequal objects via hybrid local-global optimization strategies (e.g., ITS-PUCC for circle packing), employing local minima search along with combinatorial neighborhood exploration (Ye et al., 2013).

6. Packing Minima in Learning Theory and High-Dimensional Function Spaces

Packing minima appear in learning-theoretic minimax bounds for neural networks. In overparameterized ReLU networks, packing arguments using boundary-localized atoms demonstrate the curse of dimensionality for stable minima, where the minimax mean-squared error is lower bounded by $n^{-2/(d+1)}$, reflecting exponential deterioration in input dimension (Liang et al., 25 Jun 2025). The underlying mechanism is that packing exponentially many disjoint regions at the boundary (each creating a flat minimum) leads to "neural shattering," severely limiting generalization rates for curvature-controlled solutions. This phenomenon contrasts sharply with weight-decay minima, which avoid boundary-localization and exhibit polynomial-in-$d$ rates.

7. Open Problems and Research Directions

Key unsolved questions in the theory of packing minima include:

• Establishing "perfect packing-minima transference," analogous to classical Mahler–Minkowski bounds: $$\rho_j(K, \Lambda) \cdot \rho_{n-j+1}(K^*, \Lambda^*) \geq 1/2$$, for all $j$ (Henk et al., 2020).
• Extending discrete Minkowski-type lower bounds for lattice point enumeration in general dimensions using packing minima (Henk et al., 2020).
• Discovering sharper upper bounds for volume in terms of packing minima for centered, non-symmetric convex bodies (Han et al., 20 Jan 2026).
• Understanding global pessimal shapes in packing densities beyond spheres and triangles, and resolving open questions about local minima in higher dimensions (Kallus, 2013, Kallus, 2012).
• Characterizing packing minima for recursive or container minimization combinatorial problems, and closing complexity gaps in algorithms for NP-hard packing minimization problems (Alt et al., 2016, Gleixner et al., 2017).

The development and analysis of packing minima provide a foundational approach to extremal geometry, discrete optimization, computational packing, and high-dimensional learning-theoretic minimax rates, offering a unified lens on compact arrangement questions in mathematics and applied domains.