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Geometric constructions for Steinitz-type bounds in dimension two

Jean-Christophe Pain

Abstract

We investigate inequalities for partial sums of complex numbers with bounded modulus and zero total sum, a topic referred to as "polygonal confinement". Starting from Steinitz's classical result, we provide detailed constructions yielding explicit bounds, including $\sqrt{5}$, $\sqrt{3}$, $2$, and $\sqrt{2}$, depending on geometric constraints or weighted settings. The proofs are fully detailed with step-by-step constructions of permutations, highlighting the combinatorial and geometric intuition. We conclude with conjectures on optimal universal constants and directions for future research.

Geometric constructions for Steinitz-type bounds in dimension two

Abstract

We investigate inequalities for partial sums of complex numbers with bounded modulus and zero total sum, a topic referred to as "polygonal confinement". Starting from Steinitz's classical result, we provide detailed constructions yielding explicit bounds, including , , , and , depending on geometric constraints or weighted settings. The proofs are fully detailed with step-by-step constructions of permutations, highlighting the combinatorial and geometric intuition. We conclude with conjectures on optimal universal constants and directions for future research.
Paper Structure (9 sections, 7 theorems, 72 equations)

This paper contains 9 sections, 7 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Polygonal confinement
  3. Elementary polygonal confinement ($\sqrt{5}$)
  4. Steinitz theorem in dimension 2
  5. Geometric improvements and sector confinement
  6. Sector confinement
  7. Consequences
  8. Weighted sums
  9. Conclusion

Key Result

Lemma 2.1

Let $(z_i)_{1\le i\le n}$ satisfy $|z_i|\le 1$ and $\sum_{i=1}^n z_i = 0$. Then there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that where $S_p=\sum_{k=1}^p z_{\sigma(k)}$.

Theorems & Definitions (13)

  • Lemma 2.1: polygonal confinement, bound $\sqrt{5}$
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: Steinitz, dimension $2$
  • proof
  • Theorem 3.1: sector confinement
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • ...and 3 more