Two inequalities for packing symplectic balls

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Family 343 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Packing round balls into a larger round ball sounds like elementary geometry. But symplectic packing permits the balls to deform through maps that preserve the position–momentum structure of classical mechanics. The images need not remain round, and Euclidean distances are not preserved.

This flexibility makes ordinary sphere-packing intuition unreliable. Volume is still preserved, but volume alone does not decide what fits. Gromov discovered an additional obstruction: the symplectic capacities of any two packed balls cannot exceed the target capacity when added together.

Family 343 claims that, in every real dimension at least six, these are the only obstructions for any finite collection of balls. The conclusion covers unequal sizes and arbitrarily many components. It gives a complete numerical criterion, rather than an asymptotic statement for sufficiently small or sufficiently numerous equal balls.

Dimension matters sharply. Four-dimensional symplectic ball packing has additional obstructions, and the manuscript does not claim its criterion there.

2. Background and essential definitions

What a symplectic embedding preserves

Write a point of $\mathbb C^n$ as $z_j=x_j+iy_j$. The standard symplectic form is

$$\omega_0=\sum_{j=1}^n dx_j\wedge dy_j.$$

A smooth embedding $f$ is symplectic if $f^*\omega_0=\omega_0$. It preserves the two-form, and therefore the volume form $\omega_0^n/n!$. This is a stronger condition than volume preservation but weaker than preserving lengths or the metric.

For instance, stretching one coordinate $x_j$ by a factor and shrinking its paired coordinate $y_j$ by the reciprocal factor preserves $dx_j\wedge dy_j$. Symplectic images can therefore be highly distorted.

Capacity is an area, not a radius

The manuscript uses the closed ball

$$B^{2n}(R)=\left\{z\in\mathbb C^n: \pi\sum_{j=1}^n|z_j|^2\leq R\right\}.$$

Its capacity is $R$, its Euclidean radius is $\sqrt{R/\pi}$, and its volume is

$$\operatorname{Vol}(B^{2n}(R))=\frac{R^n}{n!}.$$

Thus adding capacities is not the same as adding radii. In real dimension six, for example, volume scales as the cube of capacity.

An embedding of a closed ball means a symplectic map defined on an open neighborhood of the entire closed source. The compact images must be pairwise disjoint and lie in the interior of the target ball. These conventions leave a positive amount of room and explain why the theorem uses strict inequalities.

Two necessary restrictions

For source capacities $R_1,\ldots,R_k$, volume preservation requires

$$\sum_{i=1}^kR_i^n<R^n.$$

Gromov’s two-ball obstruction gives another condition:

$$R_i+R_j<R\qquad(i\ne j).$$

The latter can prohibit a packing even when there is ample unused volume. It reflects symplectic geometry, not the elementary measure of occupied space.

3. Development history and previous partial results

Gromov’s 1985 pseudoholomorphic-curve methods transformed symplectic embedding theory. Alongside nonsqueezing phenomena, they produced the two-ball obstruction in all dimensions and additional restrictions in dimension four. McDuff–Polterovich subsequently connected packing to algebraic geometry and blowups. Gromov’s paper, McDuff–Polterovich.

Constructive methods developed in parallel. Traynor built explicit packings, while Schlenk developed planar and higher-dimensional folding techniques. Such constructions demonstrate how much more flexible symplectic images can be than rigid round balls, but a complete criterion for arbitrary collections remained a separate challenge.

Another line of progress established packing stability. Biran proved that sufficiently many equal balls can occupy arbitrarily nearly all the volume of a closed rational symplectic four-manifold. Buse–Hind extended equal-ball stability to higher-dimensional balls, projective spaces, and closed rational symplectic manifolds. Buse–Hind–Opshtein established strong packing stability for closed symplectic four-manifolds when all capacities are sufficiently small. These are regimes in which volume is effectively the only obstruction; they do not handle every arbitrary finite size list. Buse–Hind’s higher-dimensional stability paper.

Siegel–Yao formulated the conjecture settled by the new manuscript: for balls in dimensions at least six, volume and the two-ball inequalities should suffice. Their work also studied stabilized problems, where a four-dimensional domain is multiplied by a fixed symplectic surface. Four-dimensional obstructions can survive in those products. A higher-dimensional ball is a different target, so the two conclusions are compatible. Siegel–Yao.

The new paper combines explicit geometric models with a rearrangement mechanism. Its stated advance is sufficiency for every finite list of positive real capacities, including configurations with one large ball and many smaller ones.

4. What the new manuscript claims

For all integers $n\geq3$, all finite $k\geq1$, and all positive real capacities, a symplectic embedding

$$\bigsqcup_{i=1}^kB^{2n}(R_i) \hookrightarrow\operatorname{int}B^{2n}(R)$$

exists if and only if both inequalities above hold. The disjoint-union symbol means that every source ball is embedded and their images do not meet.

For one ball, the pairwise condition is empty and the volume condition reduces to $R_1<R$. For multiple balls, neither numerical test can generally be omitted.

Normalize the target capacity to one and consider real dimension six, so $n=3$:

Source capacities	Sum of cubed capacities	Largest pair sum	Conclusion under the claim
Two balls of capacity 0.6	0.432	1.2	Impossible: pairwise obstruction
Eight balls of capacity 0.49	0.941192	0.98	A packing exists
Nine balls of capacity 0.49	1.058841	0.98	Impossible: volume obstruction

These examples also emphasize that the shapes of the images may be complicated. The theorem does not say that eight ordinary round balls of those radii fit by translations and rotations.

For $k\geq2$ equal balls of capacity $r$, the criterion becomes

$$\frac rR<\min\left\{k^{-1/n},\frac12\right\}.$$

The threshold is a supremum under the closed-source, open-target convention. The displayed equality case is not included. Arbitrary real capacities are allowed; rational parameters used inside the construction do not restrict the final theorem.

5. Why the advance matters

Importance: 5/5. If correct, the result gives a complete solution to a fundamental symplectic embedding problem throughout all higher dimensions. A geometric existence problem involving potentially complicated deformations becomes decidable from two elementary numerical tests.

The contrast with dimension four is part of its importance. Higher dimension brings flexibility, but not unlimited flexibility: the two-ball obstruction survives after other packing restrictions disappear.

The claimed construction also explains how an average resource constraint can be converted into a geometric arrangement. Total available volume is an integral condition. Actual embedding demands adequate room at each local position in a model. Bridging that difference is the main methodological contribution.

The score refers to the scope of the stated theorem, not independent verification of its construction.

6. The main idea

Express a ball as a varying area requirement

Normalize the target capacity to one. The construction uses a surface of positive genus as a base and a toric domain in $\mathbb C^{n-1}$ as a fiber. Positive genus means that the surface has at least one handle.

For the fiber, introduce moment coordinates

$$p_j=\pi|z_j|^2,\qquad j=1,\ldots,n-1.$$

A toric domain is described by allowed values of these coordinates. The remaining complex coordinate contributes an area in the base. A ball of capacity $r_i$ requires the profile

$$\left(r_i-\sum_jp_j\right)_+, \qquad x_+=\max\{x,0\}.$$

Thus it needs much base area near small fiber moments and none after the fiber has consumed its capacity budget.

Let $H(p)$ be the available base-area profile. After allocating the proposed balls, the unused profile is

$$P(p)=H(p)-\sum_i\left(r_i-\sum_jp_j\right)_+.$$

The volume inequality gives positivity of its integral. But $P$ can still be negative at individual points, so simply stacking the desired layers does not work.

Redistribute before constructing the layers

The manuscript proves a Hamiltonian comparison principle for the relevant concave profiles. Coordinate rearrangements preserve integrals and lead to a fixed-point construction with a uniform positive gap. Smooth area-preserving maps then realize the rearrangements, with errors kept below that gap.

The handle on the base surface provides two transverse annuli and a closed one-form with a nonzero period. These allow the fiber rearrangement to be inserted into an exact symplectic deformation. The uniform gap supplies disjoint layers large enough to contain the requested balls.

A relative version of Moser’s method transfers the resulting packing back to the original symplectic model while controlling its boundary. The construction deliberately begins with slightly larger auxiliary source capacities, so the final requested closed balls retain a neighborhood of definition.

Build models covering the relevant target region

The remaining issue is to find suitable surface–fiber models with nearly all the target’s available volume. The paper uses projective degenerations, normal-bundle models, and symplectic cuts. These replace a region by a more structured model while retaining the compact subsets needed for a packing.

There are three cases. If every requested capacity is below one half, a complete intersection of quadrics in projective space supplies a positive-genus curve and the required model.

If one capacity is at least one half, the pairwise inequalities make it the unique such capacity. The construction reserves a slightly larger central ball and packs the smaller balls into its exterior shell. In real dimensions at least eight, successive degenerations provide an appropriate curve model of that shell.

Real dimension six needs a separate construction. The analogous simple curve would be a conic, which has genus zero and lacks the handle used by the packing mechanism. The paper instead uses a plane cubic with two marked points. Their contributions modify the integrated area profile, and a carefully preserved circle moment permits the necessary lower cut.

Finally, the relative transfer maps move the compact packings from these models into the original target ball. Preserving the relevant moments and neighborhoods during transfer is essential: an area computation by itself would not supply the asserted embeddings.

7. Scope, limitations, and open questions

The theorem is specifically about finite collections of standard balls embedded into a standard ball in real dimension at least six. It does not classify embeddings of ellipsoids, polydisks, arbitrary symplectic domains, or products with a fixed surface.

The strict inequalities matter. The paper does not establish boundary-touching packings or embeddings at exact volume equality under the stated convention. Packings can approach a limiting threshold without attaining it as compact subsets of an open target.

The result establishes existence, not uniqueness up to isotopy, a classification of the space of packings, or a computationally efficient recipe for producing explicit coordinate formulas. The proof’s auxiliary degenerations and Hamiltonian rearrangements are more elaborate than the final criterion suggests.

Finally, this digest treats the claimed sufficiency argument as stated. It does not check the Hamiltonian comparison, the six-dimensional model, or the symplectic transfer arguments independently.

8. Sources and further reading

Siegel–Yao’s paper is the closest source for the conjecture and the distinction from stabilized packing. Gromov and the subsequent packing literature explain the rigidity and stability background. Optional terminology appears under symplectic embedding and symplectic capacity.

Source manuscripts

• M1. OpenAI, Symplectic Ball Packings in Higher Dimensions (September 23, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.