The Euclidean Steinitz-Bergstrom conjecture: is the Euclidean Steinitz constant O(sqrt d)?

Theorem 1.2: there is an absolute constant $C$ such that every zero-sum family of vectors in the Euclidean unit ball of $\mathbb R^d$ (repetitions allowed) has an ordering with all partial sums of norm at most $C\sqrt d$; with the simplex lower bound, $\tfrac12\sqrt d\le S_2(d)\le C\sqrt d$. It follows from the prescribed-order signing theorem (separate entry). The constant is not made explicit, and the construction is existential: no online rule or efficient algorithm is given.

Theorem 1.2: there is an absolute constant $C$ such that every zero-sum family of vectors in the Euclidean unit ball of $\mathbb R^d$ (repetitions allowed) has an ordering with all partial sums of norm at most $C\sqrt d$; with the simplex lower bound, $\tfrac12\sqrt d\le S_2(d)\le C\sqrt d$. It follows from the prescribed-order signing theorem (separate entry). The constant is not made explicit, and the construction is existential: no online rule or efficient algorithm is given.