Polytope Research Articles

Alicia Boole Stott’s models of sections of polytopes

Alicia Boole Stott (1860–1940) was an amateur mathematician who worked on four-dimensional geometry. She is remembered for finding all three dimensional sections of the four- dimensional polytopes (that is, the analogues of the three-dimensional Platonic solids), and for the discovery of many of the semi-regular polytopes. In this text we give a short biography of her and explain her method to calculate the three-dimensional sections of the four-dimensional polytopes. We illustrate her results by showing pictures of her original models and drawings.

A point in a $$nd$$ n d -polytope is the barycenter of $$n$$ n points in its $$d$$ d -faces

Using equivariant topology, we prove that it is always possible to find $n$ points in the $d$-dimensional faces of a $nd$-dimensional convex polytope $P$ so that their center of mass is a target point in $P$. Equivalently, the $n$-fold Minkowski sum of a $nd$-polytope's $d$-skeleton is that polytope scaled by $n$. This verifies a conjecture by Takeshi Tokuyama.

Non-Normal Very Ample Polytopes and their Holes

In this paper, we show that for given integers $h$ and $d$ with $h \geq 1$ and $d \geq 3$, there exists a non-normal very ample integral convex polytope of dimension $d$ which has exactly $h$ holes.

Complete Set of CHC Tetrahedrons

In this article, using K. W. Roeder’s Theorem, some properties of CHC (compact hyperbolic coxeter) tetrahedrons have been developed which are facilitated by the link of graph theory and combinatorics, and it has been found that there are exactly 3 CHC tetrahedrons upto symmetry. General Terms Three Dimensional Polytope, Hyperbolic Space, Roland K. W. Roeder Theorem.

Recent progress on the combinatorial diameter of polytopes and simplicial complexes

The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but some try to shed light on the question by generalizing it to simplicial complexes. In particular, we include here our recent and previously unpublished proof that the maximum diameter of arbitrary simplicial complexes is in $n^{Theta(d)}$ and we summarize the main ideas in the polymath 3 project, a web-based collective effort trying to prove an upper bound of type nd for the diameters of polyhedra and of more general objects (including, e. g., simplicial manifolds).

On r-Edge-Connected r-Regular Bricks and Braces and Inscribability

A classical result due to Steinitz states that a graph is isomorphic to the graph of some 3-dimensional polytope P if and only if it is planar and 3-connected. If a graph G is isomorphic to the graph of a 3-dimensional polytope inscribed in a sphere, it is said to be of inscribable type. The problem of determining which graphs are of inscribable type dates back to 1832 and was open until Rivin proved a characterization in terms of the existence of a strictly feasible solution to a system of linear equations and inequalities which we call sys(G), which, surprisingly, also appears in the context of the Traveling Salesman Problem. Using such a characterization, various classes of graphs of inscribable type can be described. Dillencourt and Smith gave a characterization of 3-connected 3-regular planar graphs that are of inscribable and a linear-time algorithm for recognizing such graphs. In this paper, their results are generalized to r-edge-connected r-regular graphs for odd r ≥ 3 in the context of the existence of strictly feasible solutions to sys(G). An answer to an open question raised by D. Eppstein concerning the inscribability of 4-regular graphs is also given.

Normality of 3-dimensional lattice polytopes

Abstract We construct examples of ample lattice polytope of dimension n ⩾ 3 such that the n − 2 times multiple of them are very ample but not normal. We also obtain two classes of normal 3-dimensional nonsingular lattice polytopes. One is a class of polytopes with relatively small internal polytopes. The other is that of polytopes which are the Minkowski sums with line segments.

AN ORACLE-BASED, OUTPUT-SENSITIVE ALGORITHM FOR PROJECTIONS OF RESULTANT POLYTOPES

We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex- and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is α - n - 1, where the input consists of α points in ℤn. Our approach is output-sensitive as it makes one oracle call per vertex and per facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6- and 7- dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in < 1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster.

A note on reachable set bounding for delayed systems with polytopic uncertainties

The problem of reachable set estimation for linear delayed systems subject to polytopic uncertainties is revisited in this paper. The maximal Lyapunov–Krasovskii functional, combined with the free-weighting matrix technique, is utilized to derive a refined condition for a non-ellipsoidal reachable set bound. One of our discoveries is that the number of Lyapunov matrices can be chosen to be greater than the number of vertices for the uncertain polytope. Moreover, the choice of appropriate Lyapunov–Krasovskii functional candidate and the introduction of Leibniz–Newton formula lead to decoupling of the system matrices and the Lyapunov matrices. The useful term which was ignored in our previous result is retained. These treatments bring much tighter bound of the reachable set than the existing ones. Finally, the negligence in our previous paper is pointed out.

Four Kähler moduli stabilisation in type IIB orientifolds with K3-fibred Calabi-Yau threefold compactification

We present a concrete and consistent procedure to generate one kind of non-perturbative superpotential, including the gaugino condensation corrections and poly-instanton corrections, in type IIB orientifold compactification with four Kahler Moduli. Then we use this kind of superpotential as well as the alphaprime-corrections to Kahler potential to fix all of the four Kahler moduli on a general Calabi-Yau manifold with typical K3-fibred volume form. In our construction, the considered Calabi-Yau threefolds are K3-fibred and admit at least one del Pezzo surface and one W-surface. Searching through all existing four dimensional reflexive lattice polytopes, we find 23 of them fulfilling all the requirements.

A note on the extension complexity of the knapsack polytope

We show that there are 0–1 and unbounded knapsack polytopes with super-polynomial extension complexity. More specifically, for each n∈N we exhibit 0–1 and unbounded knapsack polytopes in dimension n with extension complexity Ω(2n).

Explicit Constructions of Centrally Symmetric $$k$$ -Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric $$2$$-neighborly $$d$$-dimensional polytopes with about $$3^{d/2}\approx (1.73)^d$$ vertices and of centrally symmetric $$k$$-neighborly $$d$$-polytopes with about $$2^{{3d}/{20k^2 2^k}}$$ vertices. Using this result, we construct for a fixed $$k\ge 2$$ and arbitrarily large $$d$$ and $$N$$, a centrally symmetric $$d$$-polytope with $$N$$ vertices that has at least $$\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$$ faces of dimension $$k-1$$, where $$\gamma _2=1/\sqrt{3}\approx 0.58$$ and $$\gamma _k = 2^{-3/{20k^2 2^k}}$$ for $$k\ge 3$$. Another application is a construction of a set of $$3^{\lfloor d/2 -1\rfloor }-1$$ points in $$\mathbb R ^d$$ every two of which are strictly antipodal as well as a construction of an $$n$$-point set (for an arbitrarily large $$n$$) in $$\mathbb R ^d$$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.

Variance asymptotics for random polytopes in smooth convex bodies

Let $$K \subset \mathbb R ^d$$ be a smooth convex set and let $$\mathcal{P }_{\lambda }$$ be a Poisson point process on $$\mathbb R ^d$$ of intensity $${\lambda }$$ . The convex hull of $$\mathcal{P }_{\lambda }\cap K$$ is a random convex polytope $$K_{\lambda }$$ . As $${\lambda }\rightarrow \infty $$ , we show that the variance of the number of $$k$$ -dimensional faces of $$K_{\lambda }$$ , when properly scaled, converges to a scalar multiple of the affine surface area of $$K$$ . Similar asymptotics hold for the variance of the number of $$k$$ -dimensional faces for the convex hull of a binomial process in $$K$$ .

Erratum to: Very ample but not normal lattice polytopes

Erratum to: Very ample but not normal lattice polytopes

Toric elliptic fibrations and F-theory compactifications

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discriminant components, this description is far more accurate than previous studies. The discriminant locus and Kodaria fibers groups of all 102581 elliptic fibrations are computed. The maximal gauge group is SU(27), which would naively be in contradiction with 6-dimensional anomaly cancellation.

Families of polytopal digraphs that do not satisfy the shelling property

A polytopal digraph G(P) is an orientation of the skeleton of a convex polytope P. The possible non-degenerate pivot operations of the simplex method in solving a linear program over P can be represented as a special polytopal digraph known as an LP digraph. Presently there is no general characterization of which polytopal digraphs are LP digraphs, although four necessary properties are known: acyclicity, unique sink orientation (USO), the Holt–Klee property and the shelling property. The shelling property was introduced by Avis and Moriyama (2009), where two examples are given in d=4 dimensions of polytopal digraphs satisfying the first three properties but not the shelling property. The smaller of these examples has n=7 vertices. Avis, Miyata and Moriyama (2009) constructed for each d⩾4 and n⩾d+2, a d-polytope P with n vertices which has a polytopal digraph which is an acyclic USO that satisfies the Holt–Klee property, but does not satisfy the shelling property. The construction was based on a minimal such example, which has d=4 and n=6. In this paper we explore the shelling condition further. First we give an apparently stronger definition of the shelling property, which we then prove is equivalent to the original definition. Using this stronger condition we are able to give a more general construction of such families. In particular, we show that given any 4-dimensional polytope P with n0 vertices whose unique sink is simple, we can extend P for any d⩾4 and n⩾n0+d−4 to a d-polytope with these properties that has n vertices. Finally we investigate the strength of the shelling condition for d-crosspolytopes, for which Develin (2004) has given a complete characterization of LP orientations.

Log-concavity of complexity one Hamiltonian torus actions

Let (M,ω) be a closed 2n-dimensional symplectic manifold equipped with a Hamiltonian Tn−1-action. Then Atiyah–Guillemin–Sternberg convexity theorem implies that the image of the moment map is an (n−1)-dimensional convex polytope. In this Note, we show that the density function of the Duistermaat–Heckman measure is log-concave on the image of the moment map.

A counterexample to the Hirsch Conjecture

The Hirsch Conjecture (1957) stated that the graph of a $d$-dimensional polytope with $n$ facets cannot have (combinatorial) diameter greater than $n-d$. That is, that any two vertices of the polytope can be connected by a path of at most $n-d$ edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets which violates a certain generalization of the $d$-step conjecture of Klee and Walkup.

An algorithm for finding the vertices of the k -additive monotone core

Given a capacity, the set of dominating k -additive capacities is a convex polytope called the k -additive monotone core; thus, it is defined by its vertices. In this paper, we deal with the proble...

Toric K3-fibred Calabi-Yau manifolds with del Pezzo divisors for string compactifications

AbstractWe analyse several explicit toric examples of compact K3-fibred Calabi-Yau three-folds. These manifolds can be used for the study of string dualities and are crucial ingredients for the construction of LARGE Volume type IIB vacua with promising applications to cosmology and particle phenomenology. In order to build a phenomenologically viable model, on top of the two moduli corresponding to the base and the K3 fibre, we demand also the existence of two additional rigid divisors: the first supporting the non-perturbative effects needed to achieve moduli stabilisation, and the second allowing the presence of chiral matter on wrapped D-branes. We clarify the topology of these rigid divisors by discussing the interplay between a diagonal structure of the Calabi-Yau volume and D-terms. Del Pezzo divisors appearing in the volume form in a completely diagonal way are natural candidates for supporting non-perturbative effects and for quiver constructions, while ‘non-diagonal’ del Pezzo and rigid but not del Pezzo divisors are particularly interesting for model building in the geometric regime. Searching through the existing list of four dimensional reflexive lattice polytopes, we find 158 examples admitting a Calabi-Yau hypersurface with a K3 fibration and four Kähler moduli where at least one of the toric divisors is a ‘diagonal’ del Pezzo. We work out explicitly the topological details of a few examples showing how, in the case of simplicial polytopes, all the del Pezzo divisors are ‘diagonal’, while ‘non-diagonal’ ones appear only in the case of non-simplicial polytopes. A companion paper will use these results in the study of moduli stabilisation for globally consistent explicit Calabi-Yau compactifications with the local presence of chirality.