Refined Euler\u27s inequalities in plane geometries and spaces

Refined famous Euler\u27s 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

The Distance Matrix for a Simplex

In this paper we consider some geometric questions relating to a higher dimensional analogon of a triangle, called simplex. In particular, weshall be concerned with the distance matrix of its vertices. We shall also majorize the volume of a simplex in terms of the distances between vertices. As consequences, we shall derive some inequalities for determinants and, in particular, an improvement of the well-known Hadamard\u27s inequality. We shall also point to some possible applications to the chemical graph theory