Embedding Stacked Polytopes on a Polynomial-Size Grid

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 $O(n^{2\log(2d)})$, except for one axis, where the coordinates are bounded by $O(n^{3\log(2d)})$. 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

Small grid embeddings of 3-polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

A Quantitative Steinitz Theorem for Plane Triangulations

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation $G$ with $n$ vertices can be embedded in $\mathbb{R}^2$ in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a $4n^3 \times 8n^5 \times \zeta(n)$ integer grid, where $\zeta(n) \leq (500 n^8)^{\tau(G)}$ and $\tau(G)$ denotes the shedding diameter of $G$, a quantity defined in the paper.Comment: 25 pages, 6 postscript figure

Strictly convex drawings of planar graphs

Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The revision includes numerous small additions, corrections, and improvements, in particular: - a discussion of the constants in the O-notation, after the statement of thm.1. - a different set-up and clarification of the case distinction for Lemma

Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. We show that {\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors

Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. We show that {\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors

Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde