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arXiv · 2603.16547

Geometric constructions for Steinitz-type bounds in dimension two

Jean-Christophe Pain
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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.

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Jean-Christophe Pain. 2026-03-17. Geometric constructions for Steinitz-type bounds in dimension two. https://arxiv.org/abs/2603.16547

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