Geometry and topology · Oct 6, 2026

The Euclidean Steinitz–Bergström bound

Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within CdC\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 S2(d)=Θ(d)S_2(d)=\Theta(\sqrt d).

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