One observation is, that the coordinate functions can't be smooth in the corners of the convex hull, so for polynomials as coordinate functions, the only points on the curve that qualify as corners of the CH would be the end points of the curve segments.
That observation yields the sufficient condition, that the inner points of the curve segments must have non-negative barycentric coordinates with respect to the simplex defined by the curve segment's endpoints.
The oloid however is the convex hull of two 3D curves, that is the union of an uncountable number of line-segments without being a polyhedron.