On the Ambartzumian-Pleijel identity in hyperbolic geometry

TL;DR

This paper introduces a hyperbolic version of the Ambartzumian-Pleijel identity, enabling new proofs of fundamental geometric formulas and computations in hyperbolic geometry, with extensions to other constant curvature spaces.

Contribution

It develops a hyperbolic Ambartzumian-Pleijel identity and applies it to derive the hyperbolic Crofton formula, isoperimetric inequality, and chord length distribution, extending results to other constant curvature manifolds.

Findings

Derived the hyperbolic Crofton formula

Proved the hyperbolic isoperimetric inequality

Computed chord length distribution for ideal polygons

Abstract

We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results for a maximally symmetric, simply connected, $2$-dimensional Riemannian manifold with constant sectional curvature are given at the end.