Strengthened Euler's Inequality in Spherical and Hyperbolic Geometries
Ren Guo, Estonia Black, Caleb Smith
TL;DR
This work develops a unified framework to extend and strengthen Euler’s inequality across Euclidean, spherical, and hyperbolic geometries. By introducing the mapping $s(x)$ that encodes the respective geometry ($s(x)=\tfrac{x}{2}$ in Euclidean, $\sinh(\tfrac{x}{2})$ in hyperbolic, and $\sin(\tfrac{x}{2})$ in spherical), the authors transfer Euclidean inequalities to curved geometries and prove a general radius-ratio bound valid in all three settings, with equality at equilateral triangles. They present the Original Strengthening in Euclidean and spherical cases and show that a direct hyperbolic analogue does not exist, highlighting fundamental geometric differences. A Symmetric Strengthening is then developed for all three geometries, yielding a chain of bounds involving symmetric sums of side-length ratios (or their trigonometric/hyperbolic analogues), with equality at equilateral configurations in the Euclidean and spherical cases, while noting the absence of a direct hyperbolic counterpart. Overall, the paper broadens the Euler-type inequalities to curved spaces through a unified, geometry-aware approach and clarifies the limits of such generalizations in hyperbolic geometry.
Abstract
Euler's inequality is a well known inequality relating the inradius and circumradius of a triangle. In Euclidean geometry, this inequality takes the form $R \geq 2r$ where $R$ is the circumradius and $r$ is the inradius. In spherical geometry, the inequality takes the form $\tan(R) \geq 2\tan(r)$ as proved in \cite{MPV}; similary, we have $\tanh(R) \geq 2\tanh(r)$ for hyperbolic triangles (see \cite{SV} for proof). In Euclidean geometry, this inequality can be strengthened as discussed in \cite{SV}. We prove an analogous version of this strengthened inequality which holds in spherical geometry, as well as an additional strengthening of Euler's inequality which holds in Euclidean geometry and can be generalized into both spherical and hyperbolic geometry.