REFINED EULER’S INEQUALITIES IN PLANE GEOMETRIES AND SPACES

Abstract

Refined famous Euler’s inequalities R ≥ nr of an n-dimensional simplex for n = 2, 3 and 4 as well as of non-Euclidean triangles in terms of symmetric functions of edge lengths of a triangle or a simplex in question are shown. Here R is the circumradius and r the inradius of the simplex. We also provide an application to geometric probabilities of our results and an example from astrophysics to the position of a planet within the space of four stars. We briefly discuss a recursive algorithm to get similar inequalities in higher dimensions.