A note on the Steinitz Lemma

We establish the connection between the Steinitz problem for ordering vector families in arbitrary norms and its variant for not necessarily zero-sum families consisting of `nearly unit’ vectors.

💡 Research Summary

The paper revisits the classical Steinitz Lemma, which guarantees that any finite family of vectors in ℝⁿ of bounded norm that sums to zero can be reordered so that every partial sum stays within a ball whose radius depends only on the dimension. The authors extend this framework to arbitrary (possibly asymmetric) norms generated by convex bodies B containing the origin, and they introduce several refined constants.

Key definitions

• Steinitz constant S(B): the smallest C such that for any finite V⊂B with ΣV=0 there exists an ordering with every partial sum having norm ≤C (norm induced by B).
• Relaxed Steinitz constant S⁎(B): the same but without the zero‑sum requirement; the bound is placed on the deviation of the partial sum from the linear interpolation between 0 and the total sum.
• Asymmetry parameter ρ(B): max_{v∈B}‖−v‖_B; equals 1 for symmetric bodies.
• ε‑Steinitz constants S_ε(B) and S⁎_ε(B): the same notions as above, but only families whose vectors have norm in the interval