Family 343Differential geometryNo LeanNot yet digested

Symplectic ball packing in higher dimensions

Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension

2n\ge6

. Finitely many closed symplectic balls of capacities

R_1,\ldots,R_k

embed disjointly into an open ball of capacity R exactly when

\sum_iR_i^n\lt R^n

and

R_i+R_j\lt R

for every distinct pair i, j.

Manuscripts in this family · 1

Symplectic Ball Packings in Higher Dimensions

Sep 23, 2026·paper.pdf

We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities

R_1,\ldots,R_k

, the closed standard symplectic

2n

-balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if

\displaystyle \sum_{i=1}^k R_i^n\lt R^n, \qquad R_i+R_j\lt R\quad(i\ne j).

Capacity is π times the squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. This proves Siegel and Yao's Conjecture A.

Full abstract

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