In this paper, we are mainly concerned with
-dimensional simplices in hyperbolic space
. We will also consider simplices with ideal vertices, and we suggest that the reader keeps the Poincaré unit ball model of hyperbolic space in mind, in which the sphere at infinity
corresponds to the bounding sphere of radius 1. It is known that all hyperbolic simplices (even the ideal ones) have finite volume. However, explicit calculation of their volume is generally a very difficult problem (see, for example,  or ). Our first theorem states that, amongst all simplices in a closed geodesic ball, the simplex of maximal volume is regular. We call a simplex regular if every permutation of its vertices can be realized by an isometry of
. A corresponding result for simplices in the sphere has been proved by Böröczky .