We investigate  singularities of the convex hull of a generic compact smooth
hypersurface in four-dimensional affine space up to diffeomorphisms. It
turns out there are only two new singularities (in
comparison with the previous dimension case) which appear at separate points
of the boundary of the convex hull and are not removed by small perturbation
of the original hypersurface. The first singularity does not contain
functional, but has at least nine number moduli. A normal form which does
not contain moduli at all is found for the second singularity.