On the pseudo-volume of a hyperbolic tetrahedron

The paper is devoted to study of a hyperbolic tetrahedron in the three dimensional Lobachevcky space. The aim is to compare two notions for the hyperbolic tetrahedron: its volume and its pseudo-volume. The latter is defined as the square root of the determinate for Gram matrix formed by the edges. In 1877 Italian mathematician Enrico d'Ovidio suggested that these two notions do coincide. Later it became known that this is not true. In the same time these two volumes are equal for infinitesimal tetrahedra. The classical Servois theorem told that the volume of an Euclidean tetrahedron is one sixth of the product of screw edges times distance and sine of angle between. We show that the theorem is true for the pseudo-volume of a hyperbolic tetrahedron. As a consequence we got a hyperbolic version of the Steiner theorem for the pseudo-volume. We show also that the Steiner theorem for hyperbolic volume is not valid.