Geometric constructions for Steinitz-type bounds in dimension two

TL;DR

This paper explores geometric inequalities related to partial sums of complex numbers with bounded modulus and zero total sum, providing explicit bounds and detailed constructions to advance Steinitz-type results in two dimensions.

Contribution

It introduces explicit geometric constructions for Steinitz-type bounds, including new bounds like √5, √3, 2, and √2, and discusses conjectures on optimal constants.

Findings

Explicit bounds for partial sums under geometric constraints

Step-by-step permutation constructions demonstrating bounds

Conjectures on optimal universal constants

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.