The algorithmic Fried Potato Problem in two dimensions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917900937330688 |
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| author | Criado, Francisco Santos, Francisco |
| author_facet | Criado, Francisco Santos, Francisco |
| contents | Conway's Fried Potato Problem seeks to determine the best way to cut a convex body in $n$ parts by $n-1$ hyperplane cuts (with the restriction that the $i$-th cut only divides in two one of the parts obtained so far), in a way as to minimize the maxuimum of the inradii of the parts. It was shown by Bezdek and Bezdek that the solution is always attained by $n-1$ parallel cuts. But the algorithmic problem of finding the best direction for these parallel cuts remains.
In this note we show that for a convex $m$-gon $P$, this direction (and hence the cuts themselves) can be found in time $O(m \log^4 m)$, which improves on a quadratic algorithm proposed by Cañete-Fernández-Márquez (DMD 2022). Our algorithm first preprocesses what we call the dome (closely related to the medial axis) of $P$ using a variant of the Dobkin-Kirkpatrick hierarchy, so that linear programs in the dome and in slices of it can be solved in polylogarithmic time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The algorithmic Fried Potato Problem in two dimensions Criado, Francisco Santos, Francisco Metric Geometry Computational Geometry 52B55, 52A40, 68Q25 Conway's Fried Potato Problem seeks to determine the best way to cut a convex body in $n$ parts by $n-1$ hyperplane cuts (with the restriction that the $i$-th cut only divides in two one of the parts obtained so far), in a way as to minimize the maxuimum of the inradii of the parts. It was shown by Bezdek and Bezdek that the solution is always attained by $n-1$ parallel cuts. But the algorithmic problem of finding the best direction for these parallel cuts remains. In this note we show that for a convex $m$-gon $P$, this direction (and hence the cuts themselves) can be found in time $O(m \log^4 m)$, which improves on a quadratic algorithm proposed by Cañete-Fernández-Márquez (DMD 2022). Our algorithm first preprocesses what we call the dome (closely related to the medial axis) of $P$ using a variant of the Dobkin-Kirkpatrick hierarchy, so that linear programs in the dome and in slices of it can be solved in polylogarithmic time. |
| title | The algorithmic Fried Potato Problem in two dimensions |
| topic | Metric Geometry Computational Geometry 52B55, 52A40, 68Q25 |
| url | https://arxiv.org/abs/2501.13873 |