A note on Sylvester problem for random polytopes in a convex body

For any d-dimensional convex body K of unit volume, let Mr(K;n) 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 Mr(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 Mr(K;n) are simplices.