论文标题
中央对称凸体的同型包装
Homothetic packings of centrally symmetric convex bodies
论文作者
论文摘要
中央对称的凸体是一个凸的紧凑型组，具有非空内部装置，对原点是对称的。特别令人感兴趣的是那些既光滑又严格凸的人 - 在这里被称为常规对称物体，因为它们保留了$ d $维的欧几里得球的许多有用属性。我们证明，对于任何给定的常规对称身体$ c $，$ c $的副本包装带有随机选择的RADII，将具有$（2,2）$ - 稀疏的平面触点图。我们进一步证明，存在一套稳定的中央对称凸体$ c $，其中任何$（2,2）$ - 稀疏平面图可以实现为$ c $的无压力同型包装的触点图。
A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex -- known here as regular symmetric bodies -- since they retain many of the useful properties of the $d$-dimensional Euclidean ball. We prove that for any given regular symmetric body $C$, a homothetic packing of copies of $C$ with randomly chosen radii will have a $(2,2)$-sparse planar contact graph. We further prove that there exists a comeagre set of centrally symmetric convex bodies $C$ where any $(2,2)$-sparse planar graph can be realised as the contact graph of a stress-free homothetic packing of $C$.
