论文标题
两个足够的条件使多面体是（本地）rupert
Two Sufficient Conditions for a Polyhedron to be (Locally) Rupert
论文作者
论文摘要
给定两个尺寸相等的立方体，有可能 - 符合所有赔率 - 通过一个孔穿过一个孔，一个孔足够大，可以直通过另一个孔。该立方体的荒谬财产最初是由17世纪莱茵河的鲁珀特亲王指出的。令人惊讶的是，该立方体并不孤单：许多其他Polyhedra都有这种财产，我们称之为Rupert。表达多面体是鲁珀特的一种简洁方法是找到两个方向$ q $和该多面角的$ q'$，以便$π（q）$适合$π（q'）$，其中$π$代表正交投影到$ xy $ plane上。鉴于此方案，要钻孔$ q'$，我们可以删除$π^{ - 1}（π（q））$。 有一个开放的猜想，每个凸多面体都是rupert。针对这种猜想，我们给出了两个足够的条件，使多面体成为鲁珀特。这两种情况都要求多面体具有特别简单的方向$ Q $，我们将其更改为少量，以获取上述要求的$ q'$。当这样的小改动给出时，我们将其称为本地段落。限制本地案件可提供许多有价值的简化。在证明我们的主要定理的过程中，我们开发了这些局部段落的理论，涉及对小旋转如何在简单多面体上作用的分析。
Given two cubes of equal size, it is possible - against all odds - to bore a hole through one which is large enough to pass the other straight through. This preposterous property of the cube was first noted by Prince Rupert of the Rhine in the 17th century. Surprisingly, the cube is not alone: many other polyhedra have this property, which we call being Rupert. A concise way to express that a polyhedron is Rupert is to find two orientations $Q$ and $Q'$ of that polyhedron so that $π(Q)$ fits inside $π(Q')$, with $π$ representing the orthogonal projection onto the $xy$-plane. Given this scheme, to bore the hole in $Q'$ we can remove $π^{-1}(π(Q))$. There is an open conjecture that every convex polyhedron is Rupert. Aiming at this conjecture, we give two sufficient conditions for a polyhedron to be Rupert. Both conditions require the polyhedron to have a particularly simple orientation $Q$, which we alter by a very small amount to get $Q'$ as required above. When a passage is given by a very small alteration like this, we call it a local passage. Restricting to the local case turns out to offer many valuable simplifications. In the process of proving our main theorems, we develop a theory of these local passages, involving an analysis of how small rotations act on simple polyhedra.
