Related papers: The rings of n-dimensional polytopes
A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex…
In the spirit of the Genetics of the Regular Figures, by L. Fejes T\'oth, we prove the following theorem: If $2n$ points are selected in the $n$-dimensional Euclidean ball $B^n$ so that the smallest distance between any two of them is as…
When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d>1, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building…
The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…
We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient…
Let $P$ be a convex polytope in the Euclidean space $\E^n$. Consider the group $G_P$ generated by reflections in the facets of $P$. We say that $P$ {\it generates a reflection group $G_P$}. In the present paper, we list all Euclidean…
An abstract $n$-polytope $\mathcal{P}$ is a partially-ordered set which captures important properties of a geometric polytope, for any dimension $n$. For even $n \ge 2$, the incidences between elements in the middle two layers of the Hasse…
The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very…
The paper is devoted to perfect and almost perfect homogeneous polytopes in Euclidean spaces. We classified perfect and almost perfect polytopes among all regular polytopes and all semiregular polytopes excepting Archimedean solids and two…
Symmetric edge polytopes are a recent and well-studied family of centrally symmetric polytopes arising from graphs. In this paper, we introduce a generalization of this family to arbitrary simplicial complexes. We show how topological…
We describe explicitly the Grothendieck rings of finite-dimensional representations of the periplectic Lie superalgebras. In particular, the Grothendieck ring of the Lie supergroup $P(n)$ is isomorphic to the ring of symmetric polynomials…
For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a…
The aim of this paper is the determination of the largest $n$-dimensional polytope with $n+3$ vertices of unit diameter. This is a special case of a more general problem proposed by Graham.
For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in…
In this paper we state a full classification for Coxeter polytopes in $\mathbb{H}^{n}$ with $n+3$ facets which are non-compact and have precisely one non-simple vertex.
4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group W(H4) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary…
\emph{Minkowski rings} are certain rings of simple functions on the Euclidean space $W = {\mathbb{R}}^d$ with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set…
A polytope is called a Coxeter polytope if its dihedral angles are integer parts of $\pi$. In this paper we prove that if a non-compact Coxeter polytope of finite volume in $H^n$ has exactly $n+3$ facets then $n\le 16$. We also find an…
We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter…
Consider a lattice in a real finite dimensional vector space. Here, we are interested in the lattice polytopes, that is the convex hulls of finite subsets of the lattice. Consider the group $G$ of the affine real transformations which map…