Perfect prismatoids are lattice Delaunay polytopes

A perfect prismatoid is a convex polytope $P$ such that for every its facet $F$ the set $vert(P) \setminus vert(F)$ belongs to a supporting hyperplane $\alpha \parallel F$. We prove that every perfect prismatoid is affinely equivalent to some $0/1$-polytope of the same dimension. (And therefore every perfect prismatoid is a lattice polytope.) Moreover, we prove that every perfect prismatoid is a lattice Delaunay polytope.