Several examples of neigbourly polyhedra in co-dimension 4

Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Unti…

Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examples conclude this work.

关于5到12维右角双曲多面体理想顶点和有限顶点数量下限的研究

On lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra in dimensions from 5 to 12

研究通过几何方法确定了5维右角双曲多面体至少有3个理想顶点，7维至少有4个有限顶点，并通过递推关系扩展到12维。

Sabitov定理到任意维度多面体的推广

Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions

本文将Sabitov定理推广至任意维度$n\ge 3$，证明具有三角形2面的多面体体积满足特定多项式关系，并由此在任意维度证明了Bellows猜想及多面体连续变形体积不变性。

Notions of maximality for integral lattice-free polyhedra: the case of dimension three

Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra $\mathbb{Z}^d$-maximal. It is known that, for fixed $d$, the family $\mathbb{Z}^d$-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutt…

论双曲空间中的ADEG多面体

On ADEG-polyhedra in hyperbolic spaces

本文证明高维双曲Coxeter多面体的非零二面角存在下界，并通过对具有指定二面角的多面体进行构造性分类，完整刻画了仅含二面角π/2、π/3、π/6且无不相交面的ADEG多面体，发现一个新的9维14面多面体。

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particu…

On the number of irreducible points in polyhedra

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in $n$-dimensional polytope with radius $k$ given by a system of $m$ linear inequalities is at most $O(m^{\lfloor\frac{n}{2}\rfloor}\log^{n-1} k)$ if $n$ is fixed. Using this result we prove the hypothesis asserting that the teaching dimension in the class of threshold functions of $k$-valued logic in $n$ variables is $Θ(\log^{n-2} k)$ for any fixed $n\ge 2$.