Maximum volume polytopes inscribed in the unit sphere

In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the $d$-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with $d+2$ vertices in every dimension, and for polytopes with $d+3$ vertices in odd dimensions. For polytopes with $d+3$ vertices in even dimensions we give a partial solution.