Facets and volume of Gorenstein Fano polytopes

It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension . Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.     

h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, c∈N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes

Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements.Through this construction, we explain combinatorially the relationship between the Gelfand-Tsetlin polytopes (1950) and the Feigin-Fourier-Littelmann-Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand-Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin-Fourier-Littelmann-Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras.     

Gorenstein polytopes and their stringy E-functions

Inspired by ideas from algebraic geometry, Batyrev and the first named author have introduced the stringy E-function of a Gorenstein polytope. We prove that this a priori rational function is actually a polynomial, which is part of a conjecture of Batyrev and the first named author. The proof relies on a comparison result for the lattice point structure of a Gorenstein polytope P, a face F of P and the face of the dual Gorenstein polytope corresponding to F. In addition, we study joins of Gorenstein polytopes and introduce the notion of an irreducible Gorenstein polytope. We show how these concepts relate to the decomposition of nef-partitions.     

On locally spherical polytopes of type {5, 3, 5}

There are only finitely many locally projective regular polytopes of type {5, 3, 5}. They are covered by a locally spherical polytope whose automorphism group is J₁×J₁×L₂(19), where J₁ is the first Janko group, of order 175560, and L₂(19) is the projective special linear group of order 3420. This polytope is minimal, in the sense that any other polytope that covers all locally projective polytopes of type {5, 3, 5} must in turn cover this one.     

Coxeter polytopes with a unique pair of non-intersecting facets

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results by Kaplinskaja [I.M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Math. Notes 15 (1974) 88-91] and the second author [P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n+3 facets, Electron. J. Combin. 14 (2007), R69, 36 pp.], this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.     

Unimodality Questions for Integrally Closed Lattice Polytopes

It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart δ -vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the δ -vector of lattice parallelepipeds. This is the first nontrivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations.     

Triangulations of Cayley and Tutte polytopes

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun’s conjecture  , which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t$t$-Cayley   and t$t$-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph.     

On hyperbolic Coxeter polytopes with mutually intersecting facets

We prove that, apart from some well-known low-dimensional examples, any compact hyperbolic Coxeter polytope has a pair of disjoint facets. This is one of very few known general results concerning combinatorics of compact hyperbolic Coxeter polytopes. We also obtain a similar result for simple non-compact polytopes.     

关于多胞形一个新仿射不变量的应用

用组合极值方法导出了n维欧氏空间中关于原点对称的一个凸多胞形子类上一个新的仿射不变量(最近由Lutwak,Yang和Zhang引入)的解析表达式,并给出了其在凸多胞形Minkowski问题的一个应用．     

Smooth Fano Polytopes Arising from Finite Partially Ordered Sets

Gorenstein Fano polytopes arising from finite partially ordered sets will be introduced. Then we study the problem of which partially ordered sets yield smooth Fano polytopes.     

An analog of determinant related to Parshin—Kato theory and integer polytopes

Parshin—Kato theory involves a multilinear function of n+1 vectors in the n-dimensional vector space over the field ?/2?. The same function arises in the computation of the product in the group (?*)ⁿ of all roots of several polynomial equations with sufficiently generic Newton polytopes. We discuss this remarkable function and related geometry of integer polytopes.     

Extensions of dually bipartite regular polytopes

This paper addresses the problem of finding abstract regular polytopes with preassigned facets and preassigned last entry of the Schläfli symbol. Using C-group permutation representation (CPR) graphs we give a solution to this problem for dually bipartite regular polytopes when the last entry of the Schläfli symbol is even. This construction is related to a previous construction by Schulte that solves the problem when the entry of the Schläfli symbol is 6.     

对称多胞形的极值性质及其应用

凸多胞形现代理论的主要成就是被称之为Dehn-Sommerville关系的上界定理和下界定理,它们属于凸多胞形的经典组合理论．本文建立了关于对称凸多胞形的两个极值定理,它们可视为凸多胞形度量理论中的上界定理和下界定理,另外给出了两个极值定理的一个应用．     

Limitations of the hyperplane separation technique for bounding the extension complexity of polytopes

The hyperplane separation bound is a lower bound on the extension complexity of a polytope. It is the main tool in Rothvoß's proof of an exponential bound for the matching polytope (Rothvoß, 2017). We show that the technique is sensitive to the choice of slack matrix and does not improve upon the best known lower bounds for spanning tree and completion time polytopes when applied to their canonical slack matrices. Stronger bounds may be obtained by appropriate rescalings and redundancy.     

Open problems on k-orbit polytopes

We present 35 open problems on combinatorial, geometric and algebraic aspects of $k$-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before.     

On the facets of the secondary polytope

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope.The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.     

On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four

We prove that the Hilbert functions of Gorenstein Artin algebras R/I of embedding dimension four are unimodal provided I has a minimal generator in degree less than five. It is still an open question as to whether all Gorenstein Hilbert functions in codimension four are SI-sequences. It is not even known if they are all unimodal. In this article, we prove that Hilbert functions of all Gorenstein Artin algebras starting with (1, 4, 10, 20, h ₄,…), where h ₄ = 34 are unimodal. Combining this with previously known results, we obtain that all Gorenstein Hilbert functions (1, 4, h ₂, h ₃, h ₄,…4, 1) are unimodal if h ₄ ≤ 34.     

Perturbation of transportation polytopes

We describe a perturbation method that can be used to reduce the problem of finding the multivariate generating function (MGF) of a non-simple polytope to computing the MGF of simple polytopes. We then construct a perturbation that works for any transportation polytope. We apply this perturbation to the family of central transportation polytopes of order $kn\times n$, and obtain formulas for the MGFs of the feasible cone of each vertex of the polytope and the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn\times n$.