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, we shall 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's inequality. We shall also point to some possible applications to the chemical graph theory.