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arXiv 2011.034362020-11-06math.MG

中心对称凸体的位似堆积

Homothetic packings of centrally symmetric convex bodies

Sean Dewar

备注：: 28 pages, 2 figures
发表信息：: Geometriae Dedicata (2022)
DOI：: 10.1007/s10711-022-00675-w

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AI总结：

本文研究中心对称凸体的位似堆积问题，证明随机半径位似堆积的接触图具有(2,2)-稀疏性，并证明存在余贫集使得任意(2,2)-稀疏平面图可实现为无应力位似堆积的接触图。

摘要：

中心对称凸体是一个具有非空内部且关于原点对称的凸紧集。特别值得关注的是那些既光滑又严格凸的物体——本文称之为正则对称体——因为它们保留了许多d维欧几里得球的优良性质。我们证明，对于任意给定的正则对称体C，以随机选取的半径对C的副本进行位似堆积，其平面接触图将是(2,2)-稀疏的。我们进一步证明，存在一个中心对称凸体C的余贫集（comeagre set），使得任何(2,2)-稀疏平面图都可以实现为C的无应力位似堆积的接触图。

英文摘要：

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 $C$.

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