Large disks touching three sides of a quadrilateral

TL;DR

The article shows that every Jordan quadrilateral contains a disk whose boundary intersects the quadrilateral boundary on three sides, a sharp geometric property linked to the medial axis. The proof reduces to polygonal quadrilaterals with right-angle vertices, derives the result via a medial-axis path between opposite vertices, and extends to general quadrilaterals through convergence-from-inside arguments aided by modulus stability. It then derives a corollary providing quantitative bounds: for quadrilaterals with uniformly bounded modulus, the contained disk radius scales with the larger internal distance $s_{a}(Q)$ or $s_{b}(Q)$, controlled by Lehto–Virtanen estimates. Overall, the work offers a concise alternative proof and improved constants for a 2024 result by Chrontsios–Garitsis–Hinkkanen and underscores the medial axis as a versatile tool in complex analysis and quasiconformal mapping.

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.

Figures (5)

• Figure 1: Representation of the quad-vertices, $a$-sides and $b$-sides of a quadrilateral Q, together with its representation via conformal mapping to a rectangle.
• Figure 2: Example of a quadrilateral showing that the number 3 in Theorem \ref{['mainTheorem']} is sharp, where the figure corresponds to a crescent. The medial axis, which will be defined later in this section, is represented by the dotted curve.
• Figure 3: Approximation of any quadrilateral $Q$ from inside via polygons as in Lemma \ref{['lemma:approxQuad']}. The dashed curves represent the quadrilaterals converging from inside to $Q$, whereas the thick curve represents the polygon obtained in Lemma \ref{['lemma:approxQuad']}.
• Figure 4: Illustration of the proof of Lemma \ref{['lemma:main']}. The medial axis is the union of the dashed curves. The circle has its center in the medial axis. The simple path $\gamma$ joining $v_{1}$ and $v_{3}$ within the medial axis is represented in a lighter color.
• Figure 5: Representation of the circle obtained in Theorem \ref{['mainTheorem']} that intersects three sides of our quadrilateral $Q$.

Theorems & Definitions (14)

• Theorem 1.1
• Corollary 1.2: Theorem 1.1 in MR4735579
• Definition 2.1: Convergence from inside
• Lemma 2.2: Lemma 4.3, p.26 in LehtoVirtanen
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1