从Kneser-Poulsen猜想到球多面体
From the Kneser-Poulsen conjecture to ball-polyhedra
摘要：
Kneser（1955）和Poulsen（1954）独立地提出了一个关于有限球系的非常基本的几何问题。根据他们著名的猜想，如果在欧几里得空间中的一有限集的球被重新定位，使得每对球的中心之间的距离减小，则并集的体积（分别为，球的交叉点）减少（相应地，增加）。本文的前半部分综述了Kneser-Poulsen猜想在欧氏空间、球面空间和双曲空间中的研究现状，重点讨论了欧氏空间的Kneser-Poulsen猜想。因此，在欧几里得空间中，用离散几何的观点研究多个全等球的交的几何就显得十分自然和重要。我们称这些集合为球多面体。在本文的后半部分，我们回顾了球多面体已知的一系列基本结果。除了这篇论文明显的调查性质之外，我们想强调我们明确的意图，提出相当多的开放性问题，以激励进一步的研究。
英文摘要：
A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser-Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of this paper we survey a selection of fundamental results known on ball-polyhedra. Besides the obvious survey character of this paper we want to emphasize our definite intention to raise quite a number of open problems to motivate further research.
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