Family 097Convex and metric geometryLeanNot yet digested
The Euclidean Steinitz–Bergström bound
Proves that any finite sequence in the Euclidean unit ball of ℝ admits signs keeping every partial sum within , 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 .
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The Euclidean Steinitz–Bergström theorem
Sep 24, 2026·The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdfLean
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ has one signing for which every signed prefix has norm at most , 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, , and resolves the Euclidean Steinitz–Bergström conjecture.