For any d-dimensional convex body K of unit volume, d$\geq$2, let
M$_{r}$(K; n), r $\geq$ l, n$\geq$ d+ l, be the r-th order moment
of the volume of the convex hull of n random points from K. The paper
deals with the problem of determining maximizers of M$_{r}$ ( K;
n) in the class of all d-dimensional convex bodies of unit volume.
A method for selecting possible solutions, which is based on special
continuous movements of convex bodies, is presented. The results obtained
by this method support the conjecture that, for every r and n, the
only maximizers of M$_{r}$(K; n) are simplices.
