Simple Polytopes Research Articles

Graphs, skeleta and reconstruction of polytopes

A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Grunbaum’s 1967 book. This survey paper includes varied results and open questions by many researchers on simplicial polytopes, nearly simple polytopes, cubical polytopes, zonotopes, crosspolytopes, and Eulerian posets.

On the Reconstruction of Polytopes

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most \(d-2\) nonsimple vertices is determined by its 2-skeleton; and (3) for any \(d>3\) there are two d-polytopes with \(d-1\) nonsimple vertices, isomorphic \((d-3)\)-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.

Small covers over products of a simple polytope with a simplex

This paper proves that the number of small covers over products of a simple polytope with a $n$-simplex, up to D-J equivalence, is a polynomial in the variable $2^n$. A similar result holds for orientable small covers. We also provide a new way of computation, namely computing the finite number of representatives and interpolating polynomially. The ratio between the number of orientable small covers and the number of small covers is given. As an application, by interpolation, we determine the polynomials related to small covers and orientable small covers over products of a prism with a simplex up to D-J equivalence. A formula for calculating the number of equivariant homeomorphism classes of small covers over the product is also provided.

Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face

A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed numbers of k-gons, k ≥ 7 . Any simple polytope with only 5-, 6- and at most one 7-gon is Pogorelov. For any other prescribed numbers of k-gons, k ≥ 7 , we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from k-barrels (also known as Löbel polytopes, whose graphs are biladders on 2 k vertices) can be constructed from the 5- or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the ( 5 , 0 ) -nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6- and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons.

A Lower Bound Theorem for Centrally Symmetric Simplicial Polytopes

Stanley proved that for any centrally symmetric simplicial d-polytope P with \(d\ge 3\), \(g_2(P) \ge {d \atopwithdelims ()2}-d\). We provide a characterization of centrally symmetric simplicial d-polytopes with \(d\ge 4\) that satisfy this inequality as equality. This gives a natural generalization of the classical Lower Bound Theorem for simplicial polytopes to the setting of centrally symmetric simplicial polytopes.

Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

We introduce a novel scheme based on a blending of Fourier-Motzkin elimination (FME) and adjustable robust optimization techniques to compute the maximum volume inscribed ellipsoid (MVE) in a polytopic projection. It is well-known that deriving an explicit description of a projected polytope is NP-hard. Our approach does not require an explicit description of the projection, and can easily be generalized to find a maximally sized convex body of a polytopic projection. Our obtained MVE is an inner approximation of the projected polytope, and its center is a centralized relative interior point of the projection. Since FME may produce many redundant constraints, we apply an LP-based procedure to keep the description of the projected polytopes at its minimal size. Furthermore, we propose an upper bounding scheme to evaluate the quality of the inner approximations. We test our approach on a simple polytope and a color tube design problem, and observe that as more auxiliary variables are eliminated, our inner approximations and upper bounds converge to optimal solutions. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0763 .

Generalized Permutohedra from Probabilistic Graphical Models

A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. There is an analogous polytope for conditional independence relations coming from a regular Gaussian model, and it can be defined using multiinformation or relative entropy. For directed acyclic graphical models and also for mixed graphical models containing undirected, directed, and bidirected edges, we give a construction of this polytope, up to equivalence of normal fans, as a Minkowski sum of matroid polytopes. Finally, we apply this geometric insight to construct a new ordering-based search algorithm for causal inference via directed acyclic graphical models.

Fair Division and Generalizations of Sperner- and KKM-type Results

We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the Knaster--Kuratowski--Mazurkiewicz (KKM) theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate it to hyperplane mass partitions. We show the existence of fair cake division and rental harmony in the sense of Su even in the absence of full information. Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds. For simplicial polytopes our results turn out to be improvements over the earlier work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma. Moreover, our results extend the work of Musin on quantitative Sperner-type results for piecewise linear manifolds.

A generalized lower bound theorem for balanced manifolds

A simplicial complex of dimension $$d-1$$ is said to be balanced if its graph is d-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if $$\Delta $$ is an arbitrary balanced triangulation of any closed homology manifold of dimension $$d-1 \ge 3$$ , then $$2h_2(\Delta ) - (d-1)h_1(\Delta ) \ge 4{d \atopwithdelims ()2}(\tilde{\beta }_1(\Delta )-\tilde{\beta }_0(\Delta ))$$ , thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag $$h''$$ -vectors.

Ehrhart polynomials of 3-dimensional simple integral convex polytopes

We give an explicit formula on the Ehrhart polynomial of a 3-dimensional simple integral convex polytope by using toric geometry.

Buchstaber invariants of universal complexes

Davis and Januszkiewicz introduced (real and complex) universal complexes to give an equivalent definition of characteristic maps of simple polytopes, which now can be seen as “colorings”. The author derives an equivalent definition of Buchstaber invariants of a simplicial complex K, then interprets the difference of the real and complex Buchstaber invariants of K as the obstruction to liftings of nondegenerate simplicial maps from K to the real universal complex or the complex universal complex. It was proved by Ayzenberg that real universal complexes can not be nondegenerately mapped into complex universal complexes when dimension is 3. This paper presents that there is a nondegenerate map from 3-dimensional real universal complex to 4-dimensional complex universal complex.

Balanced generalized lower bound inequality for simplicial polytopes

A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the h-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial d-polytopes, that is simplicial d-polytopes whose underlying graphs are d-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their h-numbers.

Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes

We describe the combinatorics of three families of simple - dimensional polytopes which play an important role in various problems of algebraic topology, hyperbolic geometry, graph theory, and their applications. The first family consists of simple polytopes with at most hexagonal faces. The second family consists of Pogorelov polytopes. The third family consists of fullerenes and is the intersection of the first two. We show that in the case of fullerenes there are stronger results than for the first two. Our main tools are -belts of faces, simple partitions of a disc and the operations of transformation and connected sum.

Simple polytopes without small separators

We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least Ω(n/log3/2 n). This disproves a conjecture by Kalai from 1991/2004.

Certain generalized higher derived functors associated to quasitoric manifolds

In this paper, we show that the higher derived functors of the primitive element functor of $$E_*(M)$$ where M is a quasitoric manifold and E is a complex-orientable homology theory are independent of the torus action, implying they depend on the orbit space. This extends the results of an earlier work by both authors. By applying cosimplicial methods to coalgebras that are free modules over a commutative ring, we are able to generalize the aforementioned functors that appeared in Bousfield’s work. In addition, certain results that appeared in earlier work of Larry Smith have been generalized and applied to faithful systems to expose how and to what extent ESP sequences show up in the arguments. As an application, we are able to prove that a necessary condition for a simplicial complex dual to a simple convex polytope being rigid is an isomorphism of these derived functors in sufficiently large dimensions; this further exposes the relation between the homotopy type of the manifold, the torus action and the combinatorics of the orbit.

Directed Random Walks on Polytopes with Few Facets

Let P be a simple polytope with n−d=2, where d is the dimension and n is the number of facets. The graph of such a polytope is also called a grid. It is known that the directed random walk along the edges of P terminates after O(log2⁡n) steps, if the edges are oriented in a (pseudo-)linear fashion. We prove that the same bound holds for the more general unique sink orientations.

Skyscraper polytopes and realizations of 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 R2 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 4n3×8n5×ζ(n) integer grid, where ζ(n)≤(500n8)τ(G) and τ(G) denotes the shedding diameter of G, a quantity defined in the paper.

In-Block Controllability of Affine Systems on Polytopes

In this technical note, we introduce the study of the in-block controllability (IBC) of affine systems on polytopes. In particular, we study whether all the states in the interior of a given polytope are mutually accessible through its interior by applying uniformly bounded inputs. The proposed study formalizes controllability under state constraints, and it has strong connections to the study of controllability of piecewise affine hybrid systems. We provide easily checkable necessary conditions for IBC. Then, for an important class of polytopes, simplicial polytopes, we show that these conditions are also sufficient.

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

A family of closed manifolds is said to be cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. Cohomological rigidity is established here for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. The class of 3-dimensional combinatorial simple polytopes different from tetrahedra and without facets forming 3- and 4-belts is studied. This class includes mathematical fullerenes, that is, simple 3- polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope in admits in Lobachevsky 3-space a right-angled realisation which is unique up to isometry. Our families of smooth manifolds are associated with polytopes in the class . The first family consists of 3-dimensional small covers of polytopes in , or equivalently, hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes in . Our main result is that both families are cohomologically rigid, that is, two manifolds and from either family are diffeomorphic if and only if their cohomology rings are isomorphic. It is also proved that if and are diffeomorphic, then their corresponding polytopes and are combinatorially equivalent. These results are intertwined with classical subjects in geometry and topology such as the combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds, and invariance of Pontryagin classes. The proofs use techniques of toric topology. Bibliography: 69 titles.

Maximizing a Class of Utility Functions Over the Vertices of a Polytope

Given a polytope X, a monotone concave univariate function g, and two vectors c and d, we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c’x + g(d’x). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is 𝒩𝒫-hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives.