The numbers of faces of simplicial polytopes

In this paper is considered the problem of determining the possiblef-vectors of simplicial polytopes. A conjecture is made about the form of the sclution to this problem; it is proved in the case ofd-polytopes with at mostd+3 vertices.     

On simplicial partitions of polytopes

Mathematical Notes - We prove some general properties of prismoids, i.e., polytopes all of whose vertices lie in two parallel planes. On the basis of these properties, we obtain a nontrivial lower...     

Finding simplicial subdivisions of polytopes

Subdivisions of polytopes are described which place any chosen vertex in a (near-) minimal number of simplices. Ancillary procedures to find volumes and centroids of polytopes are indicated.     

On low-dimensional faces that high-dimensional polytopes must have

We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk1 there is an integer f(k) such that everyd-polytope,df(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube.We give some related results concerning facet-forming polytopes and tilings. For example, sharpening a result of Schulte [25] we prove that there is no face to face tiling of ⁵ with crosspolytopes.Supported in part by a BSF Grant and by I.H.E.S, Bures-Sur-Yvette.     

Gradient superconvergence on uniform simplicial partitions of polytopes

Superconvergence of the gradient for the linear simplicial finite-elementmethod applied to elliptic equations is a well known featurein one, two, and three space dimensions. In this paper we showthat, in fact, there exists an elegant proof of this featureindependent of the space dimension. As a result, superconvergencefor dimensions four and up is proved simultaneously. The keyingredient will be that we embed the gradients of the continuouspiecewise linear functions into a larger space for which wedescribe an orthonormal basis having some useful symmetry properties.Since gradients and rotations of standard finite-element functionsare in fact the rotation-free and divergence-free elements ofRaviart–Thomas and Nédélec spaces in threedimensions, we expect our results to have applications alsoin those contexts.     

Homotopy simplicial faces and the homology of realizations of simplicial topological spaces

The notion of a differential module with homotopy simplicial faces is introduced, which is a homotopy analog of the notion of a differential module with simplicial faces. The homotopy invariance of the structure of a differential module with homotopy simplicial faces is proved. Relationships between the construction of a differential module with homotopy simplicial faces and the theories of A _∞-algebras and D _∞-differential modules are found. Applications of the method of homotopy simplicial faces to describing the homology of realizations of simplicial topological spaces are presented.     

Projectivities in simplicial complexes and colorings of simple polytopes

For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group , the group of projectivities of , which is a combinatorial but not a topological invariant of . This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric manifolds. Received: 28 February 2001 / in final form: 18 May 2001/Published online: 28 February 2002     

Convex polytopes without triangular faces

LetP be a convexd-polytope without triangular 2-faces. Forj=0,…,d−1 denote byf _j(P) the number ofj-dimensional faces ofP. We prove the lower boundf _j(P)≥f _j(C _d) whereC _d is thed-cube, which has been conjectured by Y. Kupitz in 1980. We also show that for anyj equality is only attained for cubes. This result is a consequence of the far-reaching observation that such polytopes have pairs of disjoint facets. As a further application we show that there exists only one combinatorial type of such polytopes with exactly 2d+1 facets.     

Two combinatorial properties of a class of simplicial polytopes

Letf(P _s ^d ) be the set of allf-vectors of simpliciald-polytopes. ForP a simplicial 2d-polytope let Σ(P) denote the boundary complex ofP. We show that for eachf ∈f(P _s ^d ) there is a simpliciald-polytopeP withf(P)=f such that the 11 02 simplicial diameter of Σ(P) is no more thanf ₀(P)−d+1 (one greater than the conjectured Hirsch bound) and thatP admits a subdivision into a simpliciald-ball with no new vertices that satisfies the Hirsch property. Further, we demonstrate that the number of bistellar operations required to obtain Σ(P) from the boundary of ad-simplex is minimum over the class of all simplicial polytopes with the samef-vector. This polytopeP will be the one constructed to prove the sufficiency of McMullen's conditions forf-vectors of simplicial polytopes.     

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 is 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 ^Θ(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).     

Centrally symmetric polytopes with many faces

We present explicit constructions of centrally symmetric polytopes with many faces: (1) we construct a d-dimensional centrally symmetric polytope P with about 3 ^d/4 ≈ (1.316) ^d vertices such that every pair of non-antipodal vertices of P spans an edge of P, (2) for an integer k ≥ 2, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < δ _k < 1 at least (1 ? (δ _k ) ^d )( _k ^N ) k-subsets of the set of vertices span faces of P, and (3) for an integer k ≥ 2 and α > 0, we construct a centrally symmetric polytope Q with an arbitrarily large number of vertices N and of dimension d = k ^1+o(1) such that at least $(1 - k^{ - \alpha } )(_k^N )$ k-subsets of the set of vertices span faces of Q.     

Realizability and inscribability for simplicial polytopes via nonlinear optimization

We consider the problem of screening where a seller puts up for sale an indivisible good, and a buyer with a valuation unknown to the seller wishes to acquire the good. We assume that the buyer valuations are represented as discrete types drawn from some distribution, which is also unknown to the seller. The seller is averse to possible mis-specification of types distribution, and considers the unknown type density as member of an ambiguity set and seeks an optimal pricing mechanism in a worst case sense. We specify four choices for the ambiguity set and derive the optimal mechanism in each case.     

A simplicial homotopy algorithm for computing zero points on polytopes

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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.