Embedding Stacked Polytopes on a Polynomial-Size Grid

Abstract

A stacking operation adds a d-simplex on top of a facet of a simplicial d-polytope while maintaining the convexity of the polytope. A stacked d-polytope is a polytope that is obtained from a d-simplex and a series of stacking operations. We show that for a fixed d every stacked d-polytope with n vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by \(O(n^{2\log _2(2d)})\), except for one axis, where the coordinates are bounded by \(O(n^{3\log _2(2d)})\). The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.