Large disks touching three sides of a quadrilateral

TL;DR

This paper proves that every Jordan quadrilateral contains a disk touching three sides and uses this to provide a simplified proof and improved bounds for a recent complex analysis result.

Contribution

It introduces a new geometric property of Jordan quadrilaterals and enhances existing bounds with a shorter proof of a recent theorem.

Findings

Existence of a disk touching three sides of any Jordan quadrilateral

Simplified proof of a recent complex analysis theorem

Improved bounds on the main result

Abstract

We show that every Jordan quadrilateral $Q\subset\mathbb{C}$ contains a disk $D$ so that $\partial D\cap\partial Q$ contains points of three different sides of $Q$. As a consequence, together with some modulus estimates from Lehto and Virtanen, we offer a short proof of the main result obtained by Chrontsios-Garitsis and Hinkkanen in 2024 and it also improves the bounds on their result.