Generalizations of Triangle Inequalities to Spherical and Hyperbolic Geometry

TL;DR

This paper extends classical triangle inequalities involving inradius, circumradius, and side lengths from Euclidean geometry to spherical and hyperbolic geometries, including higher-dimensional simplices.

Contribution

It introduces new generalized inequalities for triangles and simplices in spherical and hyperbolic spaces, broadening the scope of classical Euclidean geometric inequalities.

Findings

Generalized Euler's inequality to spherical and hyperbolic triangles

Strengthened inequalities involving inradius and circumradius

Extended inequalities to n-dimensional simplices in spherical geometry

Abstract

Certain triangle inequalities involving the circumradius, inradius, and side lengths of a triangle are generalized to spherical and hyperbolic geometry. Examples include strengthenings of Euler's inequality, $R\geq2r$. An extension of Euler's inequality to a simplex in $n$-dimensional space is also generalized to spherical geometry.