The Euclidean Steinitz–Bergström bound. Proves that any finite sequence in the Euclidean unit ball of ℝd 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 $S_2(d)=\Theta(\sqrt d)$.
released 2026-09-24 | 2 theorems · 12 lemmas · 18 proofs · 10,429 words |
PLAY LEVEL 1 »(pdf)
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝd 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.