Perfect prismatoids are lattice Delaunay polytopes

TL;DR

This paper proves that all perfect prismatoids are affinely equivalent to 0/1-polytopes and are lattice Delaunay polytopes, revealing their structure and classification within lattice geometry.

Contribution

It establishes that perfect prismatoids are affinely equivalent to 0/1-polytopes and are lattice Delaunay polytopes, providing a new classification in lattice geometry.

Findings

Every perfect prismatoid is affinely equivalent to a 0/1-polytope.

Every perfect prismatoid is a lattice Delaunay polytope.

Perfect prismatoids are characterized by their facet and vertex structure.

Abstract

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.