论文标题
二维划界的高级多型
Higher Secondary Polytopes for Two-Dimensional Zonotopes
论文作者
论文摘要
最近，Galashin，Postnikov和Williams介绍了高级多型的概念，概括了Gelfand，Kapranov和Zelevinsky的次要多层。给定$ n $ - 点配置$ \ MATHCAL {a} $ in $ \ MATHBB {r}^{d-1} $，他们定义了一个convex $（n-d）$ - dimensional polytopes $ \widehatς_{1} {1}的家族。该多台面家族的$ 1 $ - 骨骼是某些组合配置的翻转图，它们概括了$ \ text {cons} \ mathcal {a} $的三角形。我们将注意力限制在$ d = 2 $上。首先，我们将Minkowski的$ 1 $ -Skeleton联系起来，$ \wideHatς_{k} + \wideHatς_{k-1} $与已已已已已已$ k $ - $ k $ sum的$ k $ sum of $ \ mathcal {a} $ n $ \ nation $ \ nathcal cons的“ hypertriangulation”的翻转图。其次，我们计算$ \widehatς_{k} $的直径和$ \widehatς_{k}+\widehatς_{k-1} $的直径。
Very recently, Galashin, Postnikov, and Williams introduced the notion of higher secondary polytopes, generalizing the secondary polytope of Gelfand, Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in $\mathbb{R}^{d-1}$, they define a family of convex $(n-d)$-dimensional polytopes $\widehatΣ_{1}, \ldots, \widehatΣ_{n-d}$. The $1$-skeletons of this family of polytopes are the flip graphs of certain combinatorial configurations which generalize triangulations of $\text{conv} \mathcal{A}$. We restrict our attention to $d=2$. First, we relate the $1$-skeleton of the Minkowski sum $\widehatΣ_{k} + \widehatΣ_{k-1}$ to the flip graph of "hypertriangulations" of the deleted $k$-sum of $\mathcal{A}$ when $\mathcal{A}$ consists of distinct points. Second, we compute the diameter of $\widehatΣ_{k}$ and $\widehatΣ_{k}+\widehatΣ_{k-1}$ for all $k$.
