Perfect Prismatoids are Lattice Delaunay Polytopes

A perfect prismatoid is a convex polytope $P$ such that for every its facet $F$ there exists a supporting hyperplane $\alpha\parallel F$ such that any vertex of $P$ belongs to either $F$ or $\alpha$. Perfect prismatoids concern with Kalai conjecture, that any centrally symmetric $d$-polytope $P$ has at least $3^d$ non-empty faces and any polytope with exactly $3^d$ non-empty faces is a Hanner polytope. Any Hanner polytope is a perfect prismatoid (but not vice versa). A $0/1$-polytope is a convex hull of some vertices of the $d$-dimensional unit cube. 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.) Let $\Lambda$ be a lattice in $\mathbb{R}^d$ and $D$ be a polytope inscribed in a sphere $B$. Denote a boundary of $B$ by $\partial B$ and an interior of $B$ by $int\, B$. The polytope D is a lattice Delaunay polytope if $\Lambda\cap int\, B=\varnothing$ and $D$ is a convex hull of $\Lambda\cap\partial B$. We prove that every perfect prismatoid is affinely equivalent to some lattice Delaunay polytope.