{"id":"openai-math/The-Euclidean-Steinitz-Bergstrom-theorem","pdf":"https://github.com/openai/math/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026","tags":["discrete-geometry","steinitz-constant","convex-geometry","vector-sums"],"score":0,"title":"The Euclidean Steinitz–Bergström theorem","domain":"Convex and metric geometry","family":"097","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MG","summary":"Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ d has one signing for which every signed prefix has norm at most $`C\\sqrt d`$, with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, $`S_2(d)=\\Theta(\\sqrt d)`$, and resolves the Euclidean Steinitz–Bergström conjecture.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":true,"family_title":"The Euclidean Steinitz–Bergström bound"}