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A note on the Steinitz Lemma

Gergely Ambrus, Rainie Heck

TL;DR

This note refines the Euclidean Steinitz problem by introducing the $\varepsilon$-Steinitz constant $S_\varepsilon(B_2^d)$ and showing that $S(B_2^d) \le \frac{1}{\varepsilon}\left(S_\varepsilon(B_2^d)+200\sqrt{\frac{d}{\log d}}\right)$ for all $0<\varepsilon<1$ and $d\ge 2$. The approach reduces the unrestricted problem to a restricted version involving vectors with nearly unit norm, via a constructive partition into cone-contained blocks and a bound on spherical-cap measures using $t=\sqrt{\frac{\log d}{2d}}$. Central to the argument are three technical lemmas that bound the sums of long and short vectors and control the spherical-cap measure $\sigma_t$, which together yield an additive error of order $O\big(\sqrt{d/\log d}\big)$. Consequently, progress on the restricted nearly-unit problem with such an additive term would translate into progress on the original problem, potentially achieving an $O(\sqrt{d})$ bound for the Euclidean Steinitz constant. The work connects geometric probability with vector-sum combinatorics through a structured partition and transference via the $\varepsilon$-Steinitz framework.

Abstract

We prove that the Euclidean Steinitz problem may be reduced to its restriction to ``nearly unit'' vectors at the cost of an additive $O\Big(\sqrt{\frac{d}{\log d}}\Big)$ error.

A note on the Steinitz Lemma

TL;DR

This note refines the Euclidean Steinitz problem by introducing the -Steinitz constant and showing that for all and . The approach reduces the unrestricted problem to a restricted version involving vectors with nearly unit norm, via a constructive partition into cone-contained blocks and a bound on spherical-cap measures using . Central to the argument are three technical lemmas that bound the sums of long and short vectors and control the spherical-cap measure , which together yield an additive error of order . Consequently, progress on the restricted nearly-unit problem with such an additive term would translate into progress on the original problem, potentially achieving an bound for the Euclidean Steinitz constant. The work connects geometric probability with vector-sum combinatorics through a structured partition and transference via the -Steinitz framework.

Abstract

We prove that the Euclidean Steinitz problem may be reduced to its restriction to ``nearly unit'' vectors at the cost of an additive error.
Paper Structure (6 sections, 7 theorems, 41 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 41 equations, 2 figures.

Table of Contents

  1. Framework and results
  2. History of the Steinitz Lemma
  3. Notational Conventions
  4. No short vectors
  5. Technical lemmas
  6. Acknowledgements

Key Result

Theorem 1

Given any finite family of vectors $V\subset \mathbb{R}^d$ of Euclidean norm at most 1 that sum to 0, one can order the elements of $V$ as $v_1, \ldots, v_n$ so that for every $k = 1, \ldots, n,$ where $C$ is a constant that depends only on the dimension $d$.

Figures (2)

  • Figure 1: Depiction of $C_t(u)$ and $K_t(u)$.
  • Figure 2: Lower bound on surface area integral using convexity

Theorems & Definitions (14)

  • Theorem 1: Euclidean Steinitz lemma
  • Definition 2: Steinitz constant
  • Theorem 3: The Steinitz Lemma for general norms Steinitz1913SevastyanovSteinitzgrinberg_sevastyanov_ValueOfSteinitzConstant
  • Conjecture 4
  • Definition 5: $\varepsilon$-Steinitz constant
  • Theorem 6
  • Theorem 7: Lévy-Steinitz theorem
  • proof : Proof of Theorem \ref{['general theorem']}
  • Lemma 8
  • Lemma 9
  • ...and 4 more