Cubical 4-Polytopes with Few Vertices

A cubical polytope is a convex polytope all of whose facets are conbinatorial cubes. A d-polytope Pis called almost simple if, in the graph of P, each vertex of Pis d-valent of (d+ 1)-valent. It is known that, for d> 4, all but one cubical d-polytopes with up to 2^d+1vertices are almost simple, which provides a complete enumeration of all the cubical d-polytopes with up to 2^d+1vertices. We show that this result is also true for d=4.     

The cubicald-polytopes with fewer than 2 d+1 vertices

A cubical polytope is a convex polytope of which every facet is a combinatorial cube. We give here a complete enumeration of all the cubicald-polytopes with fewer than 2 ^d+1 vertices, ford≥4.     

Upper bounds for edge-antipodal and subequilateral polytopes

A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilateral polytope in any d-dimensional normed space is bounded above by (d / 2 + 1) ^d for any d ≥ 2. The same upper bound then follows for the number of vertices of the edge-antipodal polytopes introduced by I. Talata [19]. This is a constructive improvement to the result of A. Pór (to appear) that for each dimension d there exists an upper bound f(d) for the number of vertices of an edge-antipodal d-polytopes. We also show that in d-dimensional Euclidean space the only subequilateral polytopes are equilateral simplices. This material is based upon work supported by the South African National Research Foundation under Grant number 2053752.     

Classification of terminal simplicial reflexive d-polytopes with 3d − 1 vertices

We classify terminal simplicial reflexive d-polytopes with 3d − 1 vertices. They turn out to be smooth Fano d-polytopes. When d is even there is one such polytope up to isomorphism, while there are two when d is uneven.     

Hamiltonian Submanifolds of Regular Polytopes

We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k -Hamiltonian if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called super-neighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S ²×S ². By this example all regular cases of n vertices with n<20 or, equivalently, all cases of regular d-polytopes with d≤9 are now decided.     

Lower Bounds for Cubical Pseudomanifolds

It is verified that the number of vertices in a d-dimensional cubical pseudomanifold is at least 2 ^d+1. Using Adin’s cubical h-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for some special classes of cubical spheres in higher dimensions.     

Gaps in the numbers of vertices of cubical polytopes,I

A cubical polytope is a convex polytope of which very facet is a combinatorial cube. We ask for the numbers which occur as vertex numbers ofd-dimensional cubical polytopes, and we show, as a first step, that every cubicald-polytope for evend≥4 has an even number of vertices.     

Equipartite polytopes

A polytope P with 2n vertices is called equipartite if for any partition of its vertex set into two equal-size sets V ₁ and V ₂, there is an isometry of the polytope P that maps V ₁ onto V ₂. We prove that an equipartite polytope in ℝ ^d can have at most 2d+2 vertices. We show that this bound is sharp and identify all known equipartite polytopes in ℝ ^d . We conjecture that the list is complete.     

How many cyclic subpolytopes can a non-cyclic polytope have?

Acyclic d-polytope is ad-polytope that is combinatorially equivalent to a polytope whose vertices lie on the moment curve {(t, t ², …,t ^d):t∈R}. Every subpolytope of an even-dimensional cyclic polytope is again cyclic. We show that a polytope [or neighborly polytope] withv vertices that is not cyclic has at mostd+1 [respectivelyd]d-dimensional cyclic subpolytopes withv−1 vertices, providedd is even andv≧d+5.     

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.     

The Reflexive Dimension of a Lattice Polytope

The reflexive dimension refldim(P) of a lattice polytope P is the minimal integer d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is finite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k. Received July 2, 2004     

Polytopes which are orthogonal projections of regular simplexes

We consider the polytopes which are certain orthogonal projections of k-dimensional regular simplexes in k-dimensional Euclidean space R ^k . We call such polytopes -polytopes. Every sufficiently symmetric polytope, such as a regular polytope, a quasi-regular polyhedron, etc., belongs to this class. We denote by P _m,n all n-dimensional -polytopes with m vertices. We show that there is a one-to-one correspondence between the elements of P _m,n and those of P _m,m–n–1 and that this correspondence preserves the symmetry of -polytopes. Using this duality, we determine some of the P _m,n 's. We also show that a -polytope is an orthogonal projection of a cross polytope if and only if it has central symmetry.     

A four-vertex theorem for polygons and its generalization to polytopes

A vertex v of a convex polygon P is called minimal (respectively maximal) if the circle going through v and its neighbouring vertices encloses the interior of P (respectively has no vertex of P in its interior) The main result of this paper is a generalization to the convex polytopes of R ^d of the following theorem: Every convex polygon has at least two minimal and two maximal vertices The proof uses a duality theory which translates some spherical properties of a convex polytope of R ^d into combinatorial properties of a convex polyhedron of R ^d+1.     

On edge-antipodal <Emphasis Type="Italic">d</Emphasis>-polytopes

A convex d-polytope in ℝ ^d is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.      

Norm Bounds for Ehrhart Polynomial Roots

M. Beck et al. found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1+(d+1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.     

Construction theorems for polytopes

Certain construction theorems are represented, which facilitate an inductive combinatorial construction of polytopes. That is, applying the constructions to ad-polytope withn vertices, given combinatorially, one gets many combinatoriald-polytopes—and polytopes only—withn+1 vertices. The constructions are strong enough to yield from the 4-simplex all the 1330 4-polytopes with up to 8 vertices.     

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.     

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.     

Toric Rings Arising from Cyclic Polytopes

Let d and n be positive integers with n ≥ d + 1 and 𝒫 ? ? ^d an integral cyclic polytope of dimension d with n vertices, and let K[𝒫] = K[?_≥0𝒜_𝒫] denote its associated semigroup K-algebra, where 𝒜_𝒫 = {(1, α) ∈ ? ^d+1: α ∈ 𝒫} ∩ ? ^d+1 and K is a field. In the present paper, we consider the problem when K[𝒫] is Cohen–Macaulay by discussing Serre's condition (R ₁), and we give a complete characterization when K[𝒫] is Gorenstein. Moreover, we study the normality of the other semigroup K-algebra K[Q] arising from an integral cyclic polytope, where Q is a semigroup generated by its vertices only.     

Complete subgraphs of the graphs of convex polytopes

It is shown that if three vertices of the graph G(l') of a convex 3-polytope P are chosen, then G(P) contains a refinement of the complete graph C₄ on four vertices, for which the three chosen vertices are principal (that is, correspond to vertices of C₄ in the refinement). In general, all four vertices may not be preassigned as principal. For dimensions d?4, simple (simplicial) d-polytopes are constructed whose graphs contain sets of three (four) vertices, which cannot all be principal in any refinement of C₄₊₁.