  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.
