The Euclidean Steinitz–Bergström theorem

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.