Generalizations of Triangle Inequalities to Spherical and Hyperbolic 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, $R\ge 2r$. An extension of Euler's inequality to a simplex in $n$-dimensional space is also generalized to spherical geometry.

Based on work done at 2017 Oregon State University Mathematics REU. 11 pages, 2 figures