Intrinsic metrics in polygonal domains

TL;DR

This paper investigates the relationships between hyperbolic and intrinsic metrics in convex polygonal domains, focusing on the triangular ratio metric in rectangles and exploring the connection between conformal radius and boundary distance.

Contribution

It provides new inequalities and insights into the interplay of hyperbolic and intrinsic metrics in polygonal domains, especially rectangles, and links conformal radius to boundary proximity.

Findings

Derived inequalities between hyperbolic and intrinsic metrics.

Analyzed the triangular ratio metric in rectangles.

Explored the relationship between conformal radius and boundary distance.

Abstract

We study inequalities between the hyperbolic metric and intrinsic metrics in convex polygonal domains in the complex plane. Special attention is paid to the triangular ratio metric in rectangles. A local study leads to an investigation of the relationship between the conformal radius at an arbitrary point of a planar domain and the distance of the point to the boundary.