{"id":"openai-math/Symplectic-Ball-Packings-in-Higher-Dimensions","pdf":"https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026","tags":["symplectic-geometry","ball-packing","gromov-width","siegel-yao"],"score":0,"title":"Symplectic Ball Packings in Higher Dimensions","domain":"Differential geometry","family":"343","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":false,"family_title":"Symplectic ball packing in higher dimensions"}