Perfect Prismatoids are Lattice Delaunay Polytopes

Abstract

A perfect prismatoid is a convex polytope P such that for every its facet F there exists a supporting hyperplane α k F such that any vertex of P belongs to either F or α. Perfect prismatoids concern with Kalai conjecture, that any centrally symmetric dpolytope P has at least 3d non-empty faces and any polytope with exactly 3d 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 Λ be a lattice in Rd and D be a polytope inscribed in a sphere B. Denote a boundary of B by ∂B and an interior of B by int B. The polytope D is a lattice Delaunay polytope if Λ∩int B = ∅ and D is a convex hull of Λ∩∂B. We prove that every perfect prismatoid is affinely equivalent to some lattice Delaunay polytope.

Keywords

• polytopes
• delaunay polytopes
• kalai conjecture