Ulam's Packing Conjecture

Ulam's Worst-Packing Convex Body Problem

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

For a compact convex body $K\subset{\mathbb{R}}^3$ with nonempty interior, let $\delta(K)$ be the supremal upper density of packings by congruent copies of $K$ with pairwise disjoint interiors. Rotations and translations of copies are allowed. For the Euclidean ball $B$, the conjectured comparison is $\delta(K)\geq\delta(B)$ for every such $K$. This is not restricted to lattice or translative packings. [S1] [S2]

Statement recorded in the supplied source.

Is the sphere the worst-packing convex solid? [S1]

Residual task reported by the archive.

The embedded review $2026-08-17$ records: Extend the local symmetric result to arbitrary convex bodies and global shape space, or exhibit a convex solid whose optimal packing density is below that of the ball. [S1]

Short English statement

Among all three-dimensional convex bodies, is a ball the shape whose densest congruent-copy packing occupies the smallest fraction of space?

Research attempt: affine lattice packings and an enclosing-body obstruction

Status. We prove the comparison for every ellipsoid, and give a quantitative bound from any enclosing ellipsoid. This does not cover all convex bodies. The inspected abstract of Kallus [S3] assumes origin symmetry and sufficiently small asphericity; neither hypothesis is silently removed. We use the Kepler theorem $\delta(B)=\pi/\sqrt{18}$ as an external theorem, with the primary formal-proof account [R1].

1. An explicit ball lattice and its determinant

Let $D_3=\{z\in\mathbb Z^3:z_1+z_2+z_3\text{ is even}\}$. It has index 2 in $\mathbb Z^3$, hence covolume 2, and its shortest nonzero vectors have length $\sqrt2$. Therefore $\Lambda=\sqrt2D_3$ has shortest length 2 and covolume $2(\sqrt2)^3=4\sqrt2$. Unit balls centered at its points have disjoint interiors and packing density \[\frac{\operatorname{vol}(B)}{\det\Lambda} =\frac{4\pi/3}{4\sqrt2}=\frac\pi{3\sqrt2}.\tag{1}\] Lattice density follows by counting one body per fundamental cell, with a boundary error of order $R^2$ in a radius-$R$ observation window; dividing by volume of order $R^3$ makes that error vanish.

2. Every ellipsoid inherits the comparison

Write an ellipsoid as $E=AB+b$, where $A$ is invertible. Apply the same affine map to every translated unit ball in the packing above. The resulting bodies are translates of the single ellipsoid $AB$; they are thus congruent to each other, as required by the problem. Interior disjointness is preserved by invertibility. Both body volume and lattice covolume are multiplied by $|\det A|$, so the density remains the value in $1$. It follows that \[\delta(E)\geq\frac\pi{\sqrt{18}}=\delta(B).\tag{2}\] The final equality is the named Kepler theorem; the constructed lower bound is proved directly here. We do not claim ellipsoids have this as their optimal density.

3. A bound for an arbitrary enclosed body

If $K\subseteq E=AB+b$, put a translate of $K$ inside each corresponding translate of $E$ in the preceding lattice packing. This is a legal packing because the enclosing ellipsoids have disjoint interiors. Its density is \[\delta(K)\geq\frac{\operatorname{vol}(K)}{\operatorname{vol}(E)} \frac\pi{\sqrt{18}}.\tag{3}\] This estimate is affine in its volume ratio, but it loses exactly the fraction of the enclosing cell occupied by $E\setminus K$. A strict volume deficit gives a bound strictly below the target. Thus minimizing the enclosing ellipsoid volume does not by itself prove Ulam's comparison.

For a contrasting class, any parallelepiped tiles space by translations along its edge vectors, giving density 1. The inequality $\delta(K)\leq1$ follows directly from disjoint interiors. This is a complete calculation for that class rather than evidence that every convex body has a tiling.

4. Adversarial checks and remaining gap

An arbitrary affine image of a packing using differently rotated copies of a nonspherical body need not consist of congruent copies: $AR_iK$ need not be a rotation of $AK$. Our ellipsoid construction avoids this error by starting with translates of a ball, which is rotation invariant. Consequently $2$ does not establish affine invariance of the unrestricted packing functional on all bodies.

The diagnostic verifies the covolume, shortest lattice-vector length in a containing integer box, and cancellation of determinant factors for several invertible rational diagonal and shear matrices. The finite shortest-vector check is supplemented by the exact parity argument above: a nonzero $D_3$ vector cannot have squared norm 1, and $(1,1,0)$ attains squared norm 2. No numerical optimizer is taken as proof of optimal packing. The missing step is a construction that compensates the volume-ratio loss in $3$ for general shapes, including nonsymmetric bodies far from ellipsoids.

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1226 (GEO-031), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Y. Kallus, The 3-ball is a local pessimum for packing, Advances in Mathematics 264 $2014$, 355–370. https://doi.org/10.1016/j.aim.2014.07.015. Bibliographic information from the supplied record.

• S3 Y. Kallus, The 3-ball is a local pessimum for packing, arXiv:1212.2551. https://arxiv.org/abs/1212.2551. Bibliographic information from the supplied record.

• R1 T. Hales et al., A formal proof of the Kepler conjecture (2015 preprint; 2017 publication), primary theorem used for the ball optimum. https://arxiv.org/abs/1501.02155.