Cross ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1

TL;DR

This paper employs generalized cross-ratios to establish the Ptolemaean inequality and Ptolemaeus' theorem within the boundaries of rank 1 symmetric spaces of negative curvature, extending classical geometric results.

Contribution

It introduces a novel approach using generalized cross-ratios to prove fundamental inequalities in the geometry of symmetric spaces of rank 1.

Findings

Proves the Ptolemaean inequality in this setting

Establishes Ptolemaeus' theorem for boundary points

Extends classical Euclidean results to symmetric spaces

Abstract

We use generalised cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of symmetric Riemannian spaces of rank 1 and of negative curvature.