Embedding Stacked Polytopes on a Polynomial-Size Grid

作者：Demaine Erik D^*; Schulz Andre

来源：Discrete & Computational Geometry, 2017, 57(4): 782-809.

DOI：10.1007/s00454-017-9887-6

• Stacked polytopes
• Integer realization
• Grid size
• Liftings

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 , except for one axis, where the coordinates are bounded by . 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.

• 出版日期2017-6
• 单位MIT