Related papers: On faces of quasi-arithmetic Coxeter polytopes
Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases these sets can be described as the…
We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand-Tsetlin polytopes and cones, as well as Berenstein-Zelevinsky polytopes, all of which have appeared in the representation theory of…
It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the…
In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalize the correspondence of facets of a polytope to the vertices of the dual polytope to general semi-algebraic convex…
In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper…
Several recent papers have examined a rational polyhedron $P_m$ whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing $m$. A combinatorial…
We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set,…
This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on…
Let H be a finite-dimensional quasi-Hopf algebra. We show for each quotient quasibialgebra Q of H that Q is a quasi-Hopf algebra whose dimension divides the dimension of H.
We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those…
Coxeter pointed out that a number of polytopes can be projected orthogonally into two dimensions in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Among them are the 600-cell and 120-cell in…
For all $n,\phi\in \mathbb{N}$ with $\phi\geqslant n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $\phi$ facets is shown to be bounded from above and from below by positive universal…
We classify here combinatorially rigid simple polytopes with three facets more than their dimension.
A closed quasigeodesic on a convex polyhedron is a closed curve that is locally straight outside of the vertices, where it forms an angle at most $\pi$ on both sides. While the existence of a simple closed quasigeodesic on a convex…
Certain problems in quadratic minimization can be reduced to finding the point $x$ of a polyhedron ${ P}$ that minimizes the distance $\|x-p\|$ for some $p\notin { P}$. This amounts to a search for the appropriate face $F$ of ${ P}$ for…
A basis of quasi-invariant module over invariants is explicitly constructed for the two-dimensional Coxeter systems with arbitrary multiplicities. It is proved that this basis consists of $m$-harmonic polynomials, thus the earlier results…
Let $P$ be a convex $d$-polytope and $0 \leq k \leq d-1$. In 2023, this author proved the following inequalities, resolving a question of B\'ar\'any: \[ \frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} +…
We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes (they all have at least 3^d nonempty faces) and show that the Hanner polytopes among…