Roughly speaking, the rank of a Delaunay polytope (first introduced in \cite{DGL92}) is its number of degrees of freedom. In \cite{DL}, a method for computing the rank of a Delaunay polytope $P$ using the hypermetrics related to $P$ is given. Here a simpler more efficient method, which uses affine dependencies instead of hypermetrics is given. This method is applied to classical Delaunay polytopes.
Then, we give an example of a Delaunay polytope, which does not have any affine basis.
The Folkman-Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank $r$ can be represented as a pseudosphere arrangement on the $(r-1)$-dimensional sphere $S^{r-1}$, is one of the most outstanding results in oriented matroid theory. In this paper, we provide a lower-dimensional version of the topological representation theorem for uniform matroid polytopes of rank $4$. We introduce $2$-weak configurations of points and pseudocircles ($2$-weak PPC configurations) on $S^2$ and prove that every uniform matroid polytope of rank $4$ can be represent…
The maximal rank of an abstract regular polytope for M24, the Mathieu group of degree 24, is 5. There are four such polytopes of rank 5 and in this note we describe them using Curtis's MOG. This description is then used to give an upper bound for the diameter of the chamber graphs of these polytopes.
The rank of a $d$-dimensional polytope $P$ is defined by $F-(d+1)$, where $F$ denotes the number of facets of $P$. In this paper, We focus on the toric rings of $(0,1)$-polytopes with small rank. We study their normality, the torsionfreeness of their divisor class groups and the classification of their isomorphism classes.
In this paper we show that the rank of every chiral polytope having a Suzuki group as automorphism group is $3$. This gives a positive answer to a conjecture of Isabel Hubard and Dimitri Leemans.
De Loera等人在2009年证明，当秩固定时，拟阵多面体的Ehrhart多项式可以在多项式时间内计算，当元素的数量变化时。证明这一点的一个关键是，当秩固定时，切锥的任何三角剖分中的单纯锥的数目在元素的数目上是多项式有界的。作者推测是否可以在多项式时间内计算Ehrhart多项式的基数，其中元素的数量和秩允许变化。本文证明了对于秩为$r$的一致拟阵，在$n$元素上，切锥的任意三角剖分中的单纯锥个数为$n-2 \choose r-1$ .因此，如果允许秩变化，则单纯锥的数量在$n$中呈指数增长。因此，当秩随着元素的数量而变化时，Brion-Lawrence类型的方法（例如Barvinok算法）不太可能有效地计算Ehrhart多项式。为了证明这一结果，我们给出了一个三角剖分，其中最大单形与完全二部图$K_{r,n-r}$的生成线是双射的。
In this paper we study the rank of polytopes contained in the 0-1 cube with respect to $t$-branch split cuts and $t$-dimensional lattice cuts for a fixed positive integer $t$. These inequalities are the same as split cuts when $t=1$ and generalize split cuts when $t > 1$. For polytopes contained in the $n$-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most $n$, and this bound is tight as Cornuéjols and Li gave an example with split rank $n$. All known examples with high split rank -- i.e., at least $cn$ for some positive constant $c < 1$ -- are defined by expone…
We characterize the cluster variables of skew-symmetrizable cluster algebras of rank 3 by their Newton polytopes. The Newton polytope of the cluster variable $z$ is the convex hull of the set of all $\mathbf{p}\in\mathbb{Z}^3$ such that the Laurent monomial ${\bf x}^{\mathbf{p}}$ appears with nonzero coefficient in the Laurent expansion of $z$ in the cluster ${\bf x}$. We give an explicit construction of the Newton polytope in terms of the exchange matrix and the denominator vector of the cluster variable. Along the way, we give a new proof of the fact that denominator vectors of non-initial c…
For each prime power $q\geq 5$, we construct a rank four chiral polytope that has a group $PSL(3,q)$ as automorphism group and Schläfli type $\{q-1,\frac{2(q-1)}{(3,q-1)},q-1\}$. We also construct rank five polytopes for some values of $q$ and we show that there is no chiral polytope of rank at least six having a group $PSL(3,q)$ or $PSU(3,q)$ as automorphism group.
This paper exhibits two families of unravelled abstract regular polytopes in Coxeter groups of type Bn. For one family they have rank 4 while the other family has arbitrarily large rank.
对于一个多胞形P，Chvatal闭包P'是通过同时加强所有可行不等式cx <= b（含积分c）到cx <= floor（b）而得到的.直到达到P的积分壳所需的该过程的迭代次数被称为Chvatal秩。如果P是[0，1]^n的子集，那么已知O（n^2 log n）次迭代总是足够的（Eisenbrand and and Schulz（1999）），并且有时至少需要（1+1/e-o（1））n次迭代（Pokutta and Stauffer（2011）），在下限和上界之间留下巨大的差距。我们证明了存在一个包含在0/1立方体中的多面体，其Chvatal秩为Omega（n^2），将缺口缩小到对数因子。事实上，甚至一个超线性的下界被提到作为一个开放的问题，由几个作者。我们选择的P是半随机背包多面体和单个分数顶点的凸包。主要的技术成分是联系Chvatal秩同时丢番图逼近w.r.t.定义P的法向量的L1范数。
