[论文解读] Neighborly and almost neighborly configurations, and their duals

Arnau Padrol|arXiv (Cornell University)|Apr 26, 2013

Optimization and Packing Problems参考文献 4被引用 4

一句话总结

本文引入了Gale缝合构造（Gale Sewing Construction），一种系统生成邻近多面体及其对偶定向拟阵的通用方法。通过利用字典序扩展与Gale对偶性，证明了所有Gale缝合邻近多面体的子多面体仍保持邻近性，从而通过原始单纯多面体的字典序剖分，完整刻画了此类多面体的组合结构。

ABSTRACT

This thesis presents new applications of Gale duality to the study of polytopes, point configurations and oriented matroids with extremal combinatorial properties. The first part of the thesis explores construction techniques for neighborly polytopes and oriented matroids. First, we provide a new interpretation of Shemer's classical Sewing Construction for neighborly polytopes in terms of lexicographic extensions of oriented matroids. This allows us to provide a simplified proof and to generalize it to oriented matroids in two ways: the Extended Sewing Construction and the Gale Sewing Construction. Estimating the number of polytopes constructed with the later, we can provide new lower bounds for the number of combinatorial types of neighborly polytopes that even improve the current best bounds for the number of polytopes. The combination of both new techniques also allows us to construct many non-realizable neighborly oriented matroids. The degree of a point configuration is the maximal codimension of its interior faces. The second part of the thesis presents various results on the combinatorial structure of point configurations whose degree is small compared to their dimension; specifically, those whose degree is smaller than [(d+1)/2], the degree of neighborly polytopes. The study of this problem comes motivated by Ehrhart theory, where a notion equivalent to the degree - for lattice polytopes - has been widely studied during the last years. In addition, the study of the degree is also related to the Generalized Lower Bound Theorem for simplicial polytopes, with Cayley polytopes and with Tverberg theory. Among other results, we present a complete combinatorial classification for point configurations of degree 1. Moreover, we show combinatorial restrictions for configurations of small degree in terms of the novel concepts of weak Cayley configurations and codegree decompositions.

研究动机与目标

• 开发一种系统化方法，用于构造邻近及几乎邻近的点配置及其对偶定向拟阵。
• 刻画Gale缝合邻近多面体内部子多面体的组合结构。
• 通过对偶性与扩展技术，建立Gale缝合邻近多面体的所有子多面体均为邻近多面体的结论。
• 以原始多面体的字典序剖分为基础，提供Gale缝合构造的组合描述。

提出的方法

• Gale缝合构造利用对偶拟阵的字典序扩展，从已有邻近多面体生成新的邻近多面体。
• 通过Gale对偶性实现点配置与向量配置之间的相互转换，同时保持组合结构不变。
• 该方法依赖于定向拟阵中的删除与收缩运算，以分析子多面体及其对偶。
• 采用缝合定理及其推广形式，从通用旗与子旗构造邻近多面体。
• 通过双拟阵扩展形式化构造：给定一个对偶拟阵M，新对偶为M[p][q]，其中p与q为特定的字典序扩展。
• 证明过程基于对秩的归纳法，并借助定向拟阵中商与收缩运算的引理，证明顶点删除下的封闭性。

实验结果

研究问题

• RQ1 能否证明Gale缝合邻近多面体的所有子多面体仍保持邻近性？
• RQ2 Gale缝合构造能否在原始缝合定理基础上推广，以包含非通用旗？
• RQ3 原始单纯多面体与Gale缝合构造中所得字典序剖分之间存在何种组合关系？
• RQ4 定向拟阵的字典序扩展如何对应于点配置上的几何操作？
• RQ5 在该方法构造邻近多面体时，何种条件可确保最优性与最小性？

主要发现

• 所有Gale缝合邻近多面体的子多面体均为邻近多面体，其证明基于对偶定向拟阵秩的归纳法。
• 通过删除一个顶点获得的子多面体的对偶，对应于原始对偶拟阵的商，具体为(N/ẽ)[p̃][q̃]，其中ẽ为基拟阵中对应的元素。
• 通过字典序扩展保持邻近性：若原始拟阵为邻近，则其扩展后亦为邻近。
• Gale缝合多面体的组合结构完全由原始单纯多面体的字典序剖分所描述。
• 该方法通过允许使用包含通用子旗的旗，推广了经典缝合定理，从而扩大了可构造邻近配置的范围。
• 证明依赖于一个关键引理：对定向拟阵P中的任意元素e，有P emove e的对偶同构于(N/ẽ)[p̃][q̃]，从而确保顶点删除下的封闭性。