The Euclidean Steinitz–Bergström bound

Proves that any finite sequence in the Euclidean unit ball of ℝ^d admits signs keeping every partial sum within C\sqrt d, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order S_2(d)=\Theta(\sqrt d). (Lean)

Manuscript

The Euclidean Steinitz–Bergström theorem

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026}

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.

Result 097 · Convex and metric geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.