Lipschitz constants and quadruple symmetrization by M\"obius transformations

TL;DR

This paper investigates how M"obius transformations used for symmetrization affect Euclidean geometry, quantifying distortion through Lipschitz constants in various metrics to better understand their geometric impact.

Contribution

It introduces methods to measure the geometric distortion caused by M"obius transformations via Lipschitz constants, providing new insights into their effects on Euclidean and chordal metrics.

Findings

Identifies cases where distortion can be quantified

Establishes bounds on Lipschitz constants for M"obius transformations

Provides tools for analyzing geometric symmetrization effects

Abstract

Due to the invariance properties of cross-ratio, M\"obius transformations are often used to map a set of points or other geometric object into a symmetric position to simplify a problem studied. However, when the points are mapped under a M\"obius transformation, the distortion of the Euclidean geometry is rarely considered. Here, we identify several cases where the distortion caused by this symmetrization can be measured in terms of the Lipschitz constant of the M\"obius transformation in the Euclidean or the chordal metric.