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arXiv · 2011.03436

Homothetic packings of centrally symmetric convex bodies

Sean Dewar
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Abstract

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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Sean Dewar. 2020-11-06. Homothetic packings of centrally symmetric convex bodies. https://doi.org/10.1007/s10711-022-00675-w

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