周围间隙解剖
Perpendicular dissections of space
摘要：
对于参考点的每对$(Q_i,Q_j)$和每个实数$r$，存在唯一的超平面$h \perp Q_iQ_j$，使得对于$h$中的点$P$的$d(P,Q_i)^2 - d(P,Q_j)^2 = r$。在$d$ -空间中取$n$参考点，对于每对$(Q_i,Q_j)$，取一个有限的实数集。对应的望远镜形成超平面的排列。我们探讨了一般参考点的超平面相交的半格的结构。主要定理是存在一个实的、可加的增益图（这是一个具有与每条边可逆关联的可加实数的图），它的平衡平面集具有与交半格相同的结构。我们研究的要求，泛型，这是有关的行为在无穷大，但仍然是神秘的，也，在建设规则的变化，为望远镜。我们调查了几个特定的安排，以期找到每个维度的确切数量的面。原型，安排所有中垂线，研究了良好和泰德曼，由几何投票理论的动机。我们的大多数特殊例子都是由该理论的扩展提出的，在该理论中，选民行使更精细的辨别力。在整个过程中，我们提出了许多研究问题。
英文摘要：
For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$ for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair $(Q_i,Q_j)$ a finite set of real numbers. The corresponding perpendiculars form an arrangement of hyperplanes. We explore the structure of the semilattice of intersections of the hyperplanes for generic reference points. The main theorem is that there is a real, additive gain graph (this is a graph with an additive real number associated invertibly to each edge) whose set of balanced flats has the same structure as the intersection semilattice. We examine the requirements for genericity, which are related to behavior at infinity but remain mysterious; also, variations in the construction rules for perpendiculars. We investigate several particular arrangements with a view to finding the exact numbers of faces of each dimension. The prototype, the arrangement of all perpendicular bisectors, was studied by Good and Tideman, motivated by a geometric voting theory. Most of our particular examples are suggested by extensions of that theory in which voters exercise finer discrimination. Throughout, we propose many research problems.
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