A family of neighborly polytopes

Ad-polytopeP is said to be neighborly provided each [d/2] vertices determine a face ofP. We construct a family ofd-polytopes that are dual to neighborly polytopes by means of facet splitting. We use this family to find a lower bound on the number of combinatorial types of neighborly polytopes. We also show that all members of this family satisfy the famous Hirsch conjecture. Research supported by NSF Grant # MCS-07466.     

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.     

A generalized sewing construction for polytopes

We describe a new technique for constructing convex polytopes—a generalization of Shemer’s sewing construction for simplicial neighborly polytopes that has been modified to allow the creation of nonsimplicial polytopes as well. We show that Bisztriczky’s ordinary polytopes can be constructed in this manner, and we also construct several infinite families of polytopes. We consider bounds on the flag f-vectors of 4-polytopes that can be inductively constructed by generalized sewing starting from the 4-simplex.     

Many Neighborly Polytopes and Oriented Matroids

In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of ${\big (( r+d ) ^{( \frac{r}{2}+\frac{d}{2} )^{2}}\big )}\big /{\big ({r}^{{(\frac{r}{2})}^{2}} {d}^{{(\frac{d}{2})}^{2}}{\mathrm{e}^{3\frac{r}{2}\frac{d}{2}}}\big )}$ for the number of combinatorial types of vertex-labeled neighborly polytopes in even dimension d with $r+d+1$ vertices. This improves current bounds on the number of combinatorial types of polytopes. The previous best lower bounds for the number of neighborly polytopes were found by Shemer in 1982 using a technique called the Sewing Construction. We provide a new simple proof that sewing works, and generalize it to oriented matroids in two ways: to Extended Sewing and to Gale Sewing. Our lower bound is obtained by estimating the number of polytopes that can be constructed via Gale Sewing. Combining both new techniques, we are also able to construct many non-realizable neighborly oriented matroids.     

Every Simplicial Polytope with at Most d + 4 Vertices Is a Quotient of a Neighborly Polytope

We show that every simplicial d-polytope with d+4 vertices is a quotient of a neighborly (2d+4)-polytope with 2d+8 vertices, using the technique of affine Gale diagrams. The result is extended to matroid polytopes. Received September 27, 1995.     

Neighborly cubical spheres and a cubical lower bound conjecture

Using mirrors and cyclic polytopes, we construct cubicald-spheres which are the analogs of cyclic polytopes in the sense that they have the ⌉d−1/2⌈-skeleta of cubes. The existence of these neighborly cubical spheres leads to a special case of an upper bound conjecture for cubical spheres, suggested by Kalai. We extend the same construction to show that the closed convex hull off-vectors of cubical spheres contains a cone described by Adin, as an analog to the generalized lower bound theorem for simplicial polytopes. Supported in part, respectively, by an NSF Postdoctoral Fellowship; NSF Grants DMS 9207700 and 9500581; and an NSF Planning Grant.     

Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and the analysis of deformed products such that specified faces (e.g., all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d?2)-polytope Q on n?1 vertices, we construct a deformed n-cube, whose projection to the last d coordinates yields a neighborly cubical d -polytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs) which have a projection to d-space that retains the complete $(\lfloor\tfrac{d}{2}\rfloor-1)We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and the analysis of deformed products such that specified faces (e.g., all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)-polytope Q on n−1 vertices, we construct a deformed n-cube, whose projection to the last d coordinates yields a neighborly cubical d -polytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs) which have a projection to d-space that retains the complete (?\tfracd2?-1)(\lfloor\tfrac{d}{2}\rfloor-1) -skeleton.     

Neighborly Cubical Polytopes

Neighborly cubical polytopes exist: for any n≥ d≥ 2r+2 , there is a cubical convex d -polytope C _d ⁿ whose r -skeleton is combinatorially equivalent to that of the n -dimensional cube. This solves a problem of Babson, Billera, and Chan. Kalai conjectured that the boundary of a neighborly cubical polytope C _d ⁿ maximizes the f -vector among all cubical (d-1) -spheres with 2 ⁿ vertices. While we show that this is true for polytopal spheres if n≤ d+1 , we also give a counterexample for d=4 and n=6 . Further, the existence of neighborly cubical polytopes shows that the graph of the n -dimensional cube, where n\ge5 , is ``dimensionally ambiguous' in the sense of Grünbaum. We also show that the graph of the 5 -cube is ``strongly 4 -ambiguous.' In the special case d=4 , neighborly cubical polytopes have f ₃ =(f ₀ /4) log ₂ (f ₀ /4) vertices, so the facet—vertex ratio f ₃ /f ₀ is not bounded; this solves a problem of Kalai, Perles, and Stanley studied by Jockusch. Received December 30, 1998. Online publication May 15, 2000.     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

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/log^3/2 n). This disproves a conjecture by Kalai from 1991/2004.     

Neighborly polytopes

A 2m-polytopeQ isneighborly if eachm vertices ofQ determine a face. It is shown that the combinatorial structure of a neighborly 2m-polytope determines the combinatorial structure of every subpolytope. We develop a construction of “sewing a vertex onto a polytope”, which, when applied to a neighborly 2m-polytope, yields a neighborly 2m-polytope with one more, vertex. Using this construction, we show that the numberg(2m+β,2m) of combinatorial types of neighborly 2m-polytopes with 2m+β vertices grows superexponentially as β→∞ (m≧2 fixed) and asm→∞ (β≧4 fixed).     

Neighborly maps with few vertices

A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX>−10 (i.e., genusg<6, if orientable) or, equivalently, all the neighborly maps withV<12 vertices.     

Neighborly combinatorial 3-manifolds with dihedral automorphism group

It is well known that for anyn≧5 the boundary complex of the cyclic 4-polytopeC(n, 4) is a neighborly combinatorial 3-sphere admitting a vertex transitive action of the dihedral groupD _n of order 2n. In this paper we present a similar series of neighborly combinatorial 3-manifolds withn≧9 vertices, each homeomorphic to the “3-dimensional Klein bottle”. Forn=9 andn=10 these examples have been observed. before by A. Altshuler and L. Steinberg. Moreover we give a computer-aided enumeration of all neighborly combinatorial 3-manifolds with such a symmetry and with at most 19 vertices. It turns out that there are only four other types, one with 10, 15, 17, 19 vertices. We also discuss the more general case of manifolds with cyclic automorphism groupC _n.     

Lattice-free polytopes and their diameter

A convex polytope in real Euclidean space islattice-free if it intersects some lattice in space exactly in its vertex set. Lattice-free polytopes form a large and computationally hard class, and arise in many combinatorial and algorithmic contexts. In this article, affine and combinatorial properties of such polytopes are studied. First, bounds on some invariants, such as the diameter and layer-number, are given. It is shown that the diameter of ad-dimensional lattice-free polytope isO(d ³). A bound ofO(nd+d ³) on the diameter of ad-polytope withn facets is deduced for a large class of integer polytopes. Second, Delaunay polytopes and [0, 1]-polytopes, which form major subclasses of lattice-free polytopes, are considered. It is shown that, up to affine equivalence, for anyd≥3 there are infinitely manyd-dimensional lattice-free polytopes but only finitely many Delaunay and [0, 1]-polytopes. Combinatorial-types of lattice-free polytopes are discussed, and the inclusion relations among the subclasses above are examined. It is shown that the classes of combinatorial-types of Delaunay polytopes and [0,1]-polytopes are mutually incomparable starting in dimension six, and that both are strictly contained in the class of combinatorial-types of all lattice-free polytopes. This research was supported by DIMACS—the Center for Discrete Mathematics and Theoretical Computer Science at Rutgers University.     

Quotients of Some Finite Universal Locally Projective Polytopes

   Abstract. This paper classifies the quotients of a finite and locally projective polytope of type {4,3,5} . Seventy quotients are found, including three regular polytopes, and nine other section regular polytopes which are themselves locally projective. The classification is done with the assistance of GAP, a computer system for algebraic computation. The same techniques are also applied to two finite locally projective polytopes respectively of type {3,5,3} and {5,3,5} . No nontrivial quotients of the latter are found.     

Manifold structures on abstract regular polytopes

Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS ³, connected sums of handlesS ¹ × S ² , euclidean and spherical space forms, and other examples with non-trivial fundamental group.     

The element number of the convex regular polytopes

Let Σ be a set of n-dimensional polytopes. A set Ω of n-dimensional polytopes is said to be an element set for Σ if each polytope in Σ is the union of a finite number of polytopes in Ω identified along (n − 1)-dimensional faces. In this paper, we consider the n-dimensional polytopes in general, and extend the notion of element sets to higher dimensions. In particular, we will show that in the 4-space, the element number of the six convex regular polychora is at least 2, and in the n-space (n ≥ 5), the element number is 3, unless n + 1 is a square number.     

Quotients of Some Finite Universal Locally Projective Polytopes

Abstract. This paper classifies the quotients of a finite and locally projective polytope of type {4,3,5} . Seventy quotients are found, including three regular polytopes, and nine other section regular polytopes which are themselves locally projective. The classification is done with the assistance of GAP, a computer system for algebraic computation. The same techniques are also applied to two finite locally projective polytopes respectively of type {3,5,3} and {5,3,5} . No nontrivial quotients of the latter are found.     

Fractional Power Series Solutions for Systems of Equations

Abstract. In this paper we extend the explorations in [8] to include the fractional power series expansions of k equations in d variables, where d>k . An analog of Newton's polygon construction which uses the Minkowski sum P of the Newton polytopes P ₁ ,...,P _k of the k equations is given for computing such series expansions. If the Newton polytopes of these equations are the same, then the common domains of convergence for the solutions correspond to the vertices of a certain fiber polytope Σ(P) . In general, our results suggest the existence of a ``mixed fiber polytope' of k polytopes. It is also indicated that there may be a relationship between these mixed fiber polytopes and a generalization of the discriminant, which we call the mixed discriminant.