Furkejuvvon:
| Váldodahkki: | |
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| Materiálatiipa: | Preprint |
| Almmustuhtton: |
2020
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| Fáttát: | |
| Liŋkkat: | https://arxiv.org/abs/2011.03436 |
| Fáddágilkorat: |
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| _version_ | 1866911759010365440 |
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| author | Dewar, Sean |
| author_facet | Dewar, Sean |
| contents | 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 $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_03436 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Homothetic packings of centrally symmetric convex bodies Dewar, Sean Metric Geometry 52C25 (Primary) 52C15, 52A10 (Secondary) 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 $C$. |
| title | Homothetic packings of centrally symmetric convex bodies |
| topic | Metric Geometry 52C25 (Primary) 52C15, 52A10 (Secondary) |
| url | https://arxiv.org/abs/2011.03436 |