Every prescribed-order finite sequence of vectors in the Euclidean unit ball of R^d has one choice of signs whose every signed prefix has norm at most C sqrt(d), with C independent of sequence length.
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of R^d has one choice of signs whose every signed prefix has norm at most C sqrt(d), with C independent of sequence length. Consequently every indexed zero-sum family admits an ordering with every unsigned prefix bounded by C sqrt(d).
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of R^d has one choice of signs whose every signed prefix has norm at most C sqrt(d), with C independent of sequence length. Consequently every indexed zero-sum family admits an ordering with every unsigned prefix bounded by C sqrt(d).
If correct, resolves the Euclidean Steinitz–Bergström conjecture up to the optimal square-root order and removes dependence on sequence length in prescribed-order signing.
Can Euclidean unit vectors be signed in a prescribed order with every prefix bounded by a constant times the square root of the dimension?
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