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(n2log(2d)), except for one axis, where the
coordinates are bounded by O(n3log(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.Comment: 22 pages, 10 Figure