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.
released 2026-09-23 | 1 theorem · 19 lemmas · 25 proofs · 18,322 words |
PLAY LEVEL 1 »(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.