Margaret M. Bayer

Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets

This paper generalizes the Dehn-Sommerville equations for simplicial spheres to related classes of objects. The underlying motivation is to understand the combinatorial structure of arbitrary polytopes, that is, polytopes that are not necessarily simplicial. Towards this end we determine the affine span of the extended f-vectors of d-polytopes. A polytope is the convex hull of finitely many points in R e . We will generally consider a polytope of affine dimension d to be a subset of Re; this is referred to as a d-polytope. A face of a polytope is the intersection of a supporting hyperplane with the polytope. For the most part we identify a polytope P with the abstract cell complex (or lattice) realized by the boundary of P, and write a face of P as the set of vertices of P it contains. That is, we will shorten F = c o n v { v 0 . . . . . Vk} to F={Vo,.. . ,Vk} when there is no risk of confusion. By convention, the empty set is considered a ( -1)-dimensional face, and the polytope itself is a d-dimentional face; these faces will be called improper faces of P. A polytope P is called simplicial if each of its faces, except possibly P itself, is a simplex (the convex hull of affinely independent points). We will write Y (respectively, ~ ) for the set of all (respectively, all simplicial) d-polytopes. The number of/-dimensional faces (or/-faces) of a polytope P is written f , and f(P)=(fo,fl . . . . . fd-1) is called the f-vector of P. The set off -vectors of all (simplicial) polytopes is written f ( ~ e ) (f(~d)). A certain transformation on the f-vectors of simplicial polytopes has arisen in a number of different contexts, and will play an important part here. For a d-polytope P define the h-vector h(P)=(ho, hl,...,he) b y h i = i ( 1 ) i s ( d j ) s= o d i f j 1 (here we use the convention

A new index for polytopes

A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized Dehn-Sommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual.

Combinatorial Aspects of Convex Polytopes

This chapter discusses some techniques that have been successful in analyzing the combinatorial aspects of convex polytopes. A convex polyhedron is a subset of ℝ d that is the intersection of a finite number of closed half-spaces. A bounded convex polyhedron is called a convex polytope. The following can be regarded as the fundamental theorem of convex polytopes: P ⊂ ℝ d is a polytope if and only if it is the convex hull of a finite set of points in ℝ d . This theorem and related results are foundational to the theory of linear programming duality, and one of the central themes of combinatorial optimization is to make this conversion for special polytopes related to specific programming problems. The problem of developing algorithms to convert from one description of a polytope to the other arises in mathematical programming and computational geometry. The second theorem states that the collection of all the faces of a polyhedron P , ordered by inclusion, is a lattice. This lattice is called the face lattice or boundary complex of P , and two polytopes are (combinatorially) equivalent if their face lattices are isomorphic. The third theorem states that the face lattices of P and P * are anti-isomorphic. Two polytopes with anti-isomorphic face lattices are said to be dual. Two important dual classes of d -polytopes are the class of simplicial d -polytopes and the class of simple d -polytopes.

The extended f -vectors of 4-polytopes

Abstract For P a d-dimensional convex polytope and S = {i1,…, is} ⊂ {0, 1,…, d−1}, let fs(P) be the number of chains of faces O ⊂ F1 ⊂ F2 ⊂ … ⊂ Fs ⊂ P with dim Fj = ij. By the generalized Dehn-Sommerville equations the dimension of the affine span of the extended f-vectors (fs(P))S ⊂ {0,1,2,3} as P ranges over all 4-polytopes is 4, and the extended f-vectors are determined by the values of f0, f1, f2 and f02. Six linear and four nonlinear inequalities on extended f-vectors of 4-polytopes are given. The consequences for the basic f-vector, (f0, f1, f2, f3), are derived. These include the inequality, 4f2 ⩾ 3f0 − 10 + 7f3, conjectured by Barnette.

Equidecomposable and weakly neighborly polytopes

A polytope is equidecomposable if all its triangulations have the same face numbers. For an equidecomposable polytope all minimal affine dependencies have an equal number of positive and negative coefficients. A subclass consists of the weakly neighborly polytopes, those for which every set of vertices is contained in a face of at most twice the dimension as the set. Theh-vector of every triangulation of a weakly neighborly polytope equals theh-vector of the polytope itself. Combinatorial properties of this class of polytopes are studied. Gale diagrams of weakly neighborly polytopes with few vertices are characterized in the spirit of the known Gale diagram characterization of Lawrence polytopes, a special class of weakly neighborly polytopes.

The toric ℎ-vectors of partially ordered sets

First published in TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 352(10), published by the American Mathematical Society.

Flag Vectors of Eulerian Partially Ordered Sets

The closed cone of flag vectors of Eulerian partially ordered sets is studied. A new family of linear inequalities valid for Eulerian flag vectors is given. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise to extreme rays of the cone for Eulerian posets. Other extreme posets are formed from consideration of the cd -index. The cone of Eulerian flag vectors is completely determined up through rank seven.

Discriminantal Arrangements, Fiber Polytopes and Formality

Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3–formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.

FACE NUMBERS AND SUBDIVISIONS OF CONVEX POLYTOPES

The first part of the paper surveys results on f-vectors, flag vectors and h-vectors of convex polytopes. These are combinatorial parameters that have been characterized for simplicial polytopes. Many of the results known in the general case depend on the connection between convex polytopes and toric varieties. The second half of the paper looks at polyhedral subdivisions of convex polytopes. The effect of subdivision on the h-vector is studied. The paper discusses the secondary polytope, which encodes the regular subdivisions of a polytope. Fiber zonotopes and the corresponding hyperplane arrangements, called discriminantal arrangements, are studied.

A Combinatorial Study of Multiplexes and Ordinary Polytopes

Bisztriczky defines a multiplex as a generalization of a simplex, and an ordinary polytope as a generalization of a cyclic polytope. This paper presents results concerning the combinatorics of multiplexes and ordinary polytopes. The flag vector of the multiplex is computed, and shown to equal the flag vector of a many-folded pyramid over a polygon. Multiplexes, but not other ordinary polytopes, are shown to be elementary. It is shown that all complete subgraphs of the graph of a multiplex determine faces of the multiplex. The toric h -vectors of the ordinary five-dimensional polytopes are given. Graphs of ordinary polytopes are studied. Their chromatic numbers and diameters are computed, and they are shown to be Hamiltonian.