Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -- in the case of a full rank Jacobian -- proves that the realization space is a manifold of dimension NG(P):=d(f_0+f_{d-1})-f_{0,d-1}, which is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3),and Robertson claimed this to be true for all polytopes, Mnev's (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general.
In this paper we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count (2) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.
Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V$. This implies that the realization space of a $4$-polytope can have the homotopy type of an arbitrary f…
The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.
Peter McMullen has developed a theory of realizations of abstract regular polytopes, and has shown that the realizations up to congruence form a pointed convex cone which is the direct product of certain irreducible subcones. We show that each of these subcones is isomorphic to a set of positive semi-definite hermitian matrices of dimension $m$ over either the real numbers, the complex numbers or the quaternions. In particular, we correct an erroneous computation of the dimension of these subcones by McMullen and Monson. We show that the automorphism group of an abstract regular polytope can h…
This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindstöm. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses th…
There are two main thrusts in the theory of regular and chiral polytopes: the abstract, purely combinatorial aspect, and the geometric one of realizations. This brief survey concentrates on the latter. The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank, while that of a chiral polytope must strictly exceed the rank. There are similar restrictions on the dimensions of realizations of regular and chiral apeirotopes. From the viewpoint of realizations in a fixed dimension, the problems are now completely solved in up to…
This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A k-triangulation is a maximal set of chords of the convex n-gon such that no k+1 of them mutually cross. We propose a combinatorial and geometric study of multitriangulations based on their stars, which play the same role as triangles of triangulations. This study leads to interpret multitriangulations by duality as pseudoline arrangements with contact points cove…
We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its f-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively u…
We study the problem of how to obtain an integer realization of a 3d polytope when an integer realization of its dual polytope is given. We focus on grid embeddings with small coordinates and develop novel techniques based on Colin de Verdière matrices and the Maxwell-Cremona lifting method. We show that every truncated 3d polytope with n vertices can be realized on a grid of size O(n^{9log(6)+1}). Moreover, for every simplicial 3d polytope with n vertices with maximal vertex degree Δ and vertices placed on an L x L x L grid, a dual polytope can be realized on an integer grid of size O(n L^{3Δ…
We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a dire…
We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.
Taking up a suggestion of David Gale from 1956, we generate sets of combinatorially isomorphic polytopes by choosing their Gale diagrams at random. We find that in high dimensions, and under suitable assumptions on the growth of the involved parameters, the obtained polytopes have strong neighborliness properties, with high probability.
We review three realizations of the associahedron that arise as secondary polytopes, from cluster algebras, and as Minkowski sums of simplices, and show that under any choice of parameters, the resulting associahedra are affinely non-equivalent.
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension $d\ge 4$, edge-transitivity of convex polytopes implies vertex-transitivity. More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such po…
We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$. Moreover, we prove a novel necessary condition on reduced polytopes in three-dimensional Euclidean space.
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 (Period. Math. Hungar. 38 (1999), 231--246). This is a constructive impr…
The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is investigated for $r\in\left(1-\frac{1}{n},1\right]$ in dimensions $\geq2$ and for $r\in\left(\frac{1}{2},1\right]$ in dimension $3$. For the $n$-dimensional cross-polytope, $2n$ points can be placed when $r\in\left(1-\frac{1}{n},1\right]$. For the three-dimensional cross-polytope, $10$ and $12$ points can be placed if and only if $r\in\left(\frac{3}{5},\frac{2}{3}\right]$ and $r\in\left(\frac{4}{7},\frac{3}{5}\right]$ respectively, an…
By correcting an example by Polyanskii, we show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$.
A lattice Delaunay polytope is known as perfect if the only ellipsoid, that can be circumscribed about it, is its Delaunay sphere. Perfect Delaunay polytopes are in one-to-one correspondence with arithmetic equivalence classes of positive quadratic functions on the n-dimensional integral lattice that can be recovered, up to a scale factor, from the representations of its minimum. We develop a structural theory of such polytopes and describe all known perfect Delaunay polytopes in dimensions one through eight. We suspect that this list is complete.
In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the $d$-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with $d+2$ vertices in every dimension, and for polytopes with $d+3$ vertices in odd dimensions. For polytopes with $d+3$ vertices in even dimensions we give a partial solution.
For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new ob…
A Delaunay polytope $P$ is said to be {\em extreme} if the only (up to isometries) affine bijective transformations $f$ of $\R^n$, for which $f(P)$ is again a Delaunay polytope, are the homotheties. This notion was introduced in \cite{DGL92}; also some examples in dimension 1, 6, 7, 15, 16, 22, 23 were constructed and it was proved that in dimension less than 6 there are no extreme Delaunay polytopes, except the segment. In this note, for every $n\geq 6$ we build an extreme Delaunay polytope $ED_n$ of dimension $n$.
We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 0< j <k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least (c_j(d)+o(1)) {n \choose j+1} for some c_j(d)>0 and at most (1-2^{-d}+o(1)){n \choose j+1}…
We construct a hollow lattice polytope (resp. a hollow lattice simplex) of dimension $14$ (resp.$~404$) and of width $15$ (resp.$~408$). They are the first known hollow lattice polytopes of width larger than dimension. We also construct a hollow (non-lattice) tetrahedron of width $2+\sqrt2$ and conjecture that this is the maximum width among $3$-dimensional hollow convex bodies. We show that the maximum lattice width grows (at least) additively with $d$. In particular, the constructions above imply the existence of hollow lattice polytopes (resp. hollow simplices) of arbitrarily large dimensio…
Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that thi…
We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and $5$. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. In dimensions $4$ and $5$, there are $348$ and $51$ polytopes, respectively, yielding many new examples for further study. We furthermore provide new upper bounds on the dimension $d$ of compact hyperbolic Coxeter polytopes with $d+k$ facets for $…
We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lannér, Kaplinskaja, Esselmann, and the second author, this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.
