Complex cross--ratios and the Ptolemaean inequality

TL;DR

This paper employs Korányi–Reimann complex cross-ratios to establish the Ptolemaean inequality and Ptolemaeus' theorem within the boundary of complex hyperbolic space and the Heisenberg group.

Contribution

It introduces a novel application of complex cross-ratios to prove classical geometric inequalities in complex hyperbolic geometry.

Findings

Proof of Ptolemaean inequality in complex hyperbolic boundary

Verification of Ptolemaeus' theorem in the Heisenberg group

Extension of classical geometry results to complex hyperbolic setting

Abstract

We use Kor\'anyi--Reimann complex cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of complex hyperbolic space and the first Heisenberg group.