Extensions of regular polytopes with preassigned Schläfli symbol

We say that a (d+1)-polytope P is an extension of a polytope K if the facets or the vertex figures of P are isomorphic to K. The Schläfli symbol of any regular extension of a regular polytope is determined except for its first or last entry. For any regular polytope K we construct regular extensions with any even number as first entry of the Schläfli symbol. These extensions are lattices if K is a lattice. Moreover, using the so-called CPR graphs we provide a more general way of constructing extensions of polytopes.     

Regular polytopes of type {4,4,3} and {4,4,4}

Abstract regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schläfli types {4,4,3} and {4,4,4}.     

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.     

Regular 4-polytopes from the Livingstone graph of Janko’s first group

The Janko group J ₁ has, up to duality, exactly two regular rank four polytopes, of respective Schl?fli types {5,3,5} and {5,6,5}. The aim of this paper is to give geometric constructions of these two polytopes, starting from the Livingstone graph.     

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.     

Variation of Curvatures of Subanalytic Spaces and Schläfli-Type Formulas

We prove a strong variational formula for Lipschitz–Killing curvaturesof subanalytic sets. As corollaries, we reprove the Chern–Gauss–Bonnettheorem and higher Schläfli formulas. The proof of the variationalformula uses normal cycles of subanalytic sets and a new method allowinga reduction from the difficult singular geometry to computations withdifferential forms.     

Forest covers and a polyhedral intersection theorem

Aforest cover of a graph is a spanning forest for which each component has at least two nodes. We consider the convex hull of incidence vectors of forest covers in a graph and show that this polyhedron is the intersection of the forest polytope and the cover polytope. This polytope has both the spanning tree and perfect matching polytopes as faces. Further, the forest cover polytope remains integral with the addition of the constraint requiring that, for some integerk, exactlyk edges be used in the solution.Research done while thae authors were visiting the Institut für Ökonometrie und Operations Research, Universität Bonn, West Germany.Financial support provided by the Natural Sciences and Engineering Research Council, Canada and the German Research Association (Deutsche Forschungsgemeneinschaft, SFB 303).     

Normal polytopes: Between discrete,continuous, and random

The first three sections of this survey represent an updated and much expanded version of the abstract of my talk at FPSAC'2010: new results are incorporated and several concrete conjectures on the interactions between the three perspectives on normal polytopes in the title are proposed. The last section outlines new challenges in general convex polytopes, motivated by the study of normal polytopes.     

Dissecting orthoschemes into orthoschemes

We exhibit a dissection, with one degree of freedom, of an arbitrary orthoscheme in Euclidean, spherical or hyperbolic d-space into d+1 orthoschemes (Section 2); this can be interpreted as a set of relations in the scissors congruence group or, weaker, as a set of functional equations for the volume. Besides special cases where the dissection is into mutually congruent parts, we obtain, in the spherical case and for a special value of the parameter, scissors congruence formulae similar to Schläfli's period formulae for the spherical orthoscheme volume (see Section 5). In Section 6 we use the dissection to explain the structure of the volume formula for asymptotic hyperbolic 3-orthoschemes due to Lobachevsky. Finally, in Section 7, by exploiting symmetries, we show that two systems of special volume relations of Schläfli (in spherical d-space) and Coxeter (for all three geometries in dimension 3) hold even on the level of dissection. In particular, it seems that all the presently known exact values for the volume of special spherical 3-simplexes hold, independently of Schläfli's differential formula, as consequences of scissors congruence relations.     

An Atlas of Small Regular Abstract Polytopes

Summary This article announces the creation of an atlas of small regular abstract polytopes. The atlas contains information about all regular abstract polytopes whose automorphism group has order 2000 or less, except those of order <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>512k$ where $k\geq1$. The article explains also the techniques used to create the atlas, and gives some summary tables. At the time of printing, the url for the atlas is http://<a href = "http://www.abstract-polytopes.com/atlas">www.abstract-polytopes.com/atlas</a>.     

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.     

Enumerating a subset of the integer points inside a Minkowski sum

Sparse elimination exploits the structure of algebraic equations in order to obtain tighter bounds on the number of roots and better complexity in numerically approximating them. The model of sparsity is of combinatorial nature, thus leading to certain problems in general-dimensional convex geometry. This work addresses one such problem, namely the computation of a certain subset of integer points in the interior of integer convex polytopes. These polytopes are Minkowski sums, but avoiding their explicit construction is precisely one of the main features of the algorithm. Complexity bounds for our algorithm are derived under certain hypotheses, in terms of output-size and the sparsity parameters. A public domain implementation is described and its performance studied. Linear optimization lies at the inner loop of the algorithm, hence we analyze the structure of the linear programs and compare different implementations.     

Mixing chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A ??mixing?? construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral.     

Mixing regular convex polytopes

The mixing operation for abstract polytopes gives a natural way to construct a minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope, and completely determining the structure of the mix in each case.     

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 $kn\times n$.     

The partition problem

In this paper we describe several forms of thek-partition problem and give integer programming formulations of each case. The dimension of the associated polytopes and some basic facets are identified. We also give several valid and facet defining inequalities for each of the polytopes.Partial Support from NSF Grants DMS 8606188 and ECS 8800281 is gratefully acknowledged.     

Ricci Positive Metrics on the Moment-Angle Manifolds

In this paper, the authors consider the problem of which (generalized) moment-angle manifolds admit Ricci positive metrics. For a simple polytope $P$, the authors can cut off one vertex $v$ of $P$ to get another simple polytope $P_{v}$, and prove that if the generalized moment-angle manifold corresponding to $P$ admits a Ricci positive metric, the generalized moment-angle manifold corresponding to $P_{v}$ also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.     

Representing the sporadic Archimedean polyhedra as abstract polytopes

We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. The information is used to characterize which of these polyhedra have acoptic Petrie schemes (that is, have well-defined Petrie duals).