The minimum number of peeling sequences of a point set
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916479625068544 |
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| author | Simon, Dániel Gábor |
| author_facet | Simon, Dániel Gábor |
| contents | Let $P$ be a set of $n$ points in $\mathbb{R}^d$, in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let $g_d(P)$ denote the number of different removal orders we can attain while erasing all points of $P$ this way, and let $g_d(n)$ be the \emph{minimum} of $g_d(P)$ over all $n$-element point sets $P\subset \mathbb{R}^d$. Dumitrescu and Tóth showed that $g_d(n)=(d+1)^{(d+1)^2n}$. We substantially improve their bound, by proving that $g_d(n)= O((d+d\ln{(d)})^{(2+\frac{(d-1)}{\lfloor d\ln{d}\rfloor})n})$. It follows that, for any $ε>0$, there exist sufficiently high dimensional point sets $P\subset \mathbb{R}^d$ with $g_d(P)\leq O(d^{(2+ε)n})$. This almost closes the gap between the upper bound and the best-known lower bound $(d+1)^n$ for large values of $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00244 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The minimum number of peeling sequences of a point set Simon, Dániel Gábor Combinatorics 52C45 Let $P$ be a set of $n$ points in $\mathbb{R}^d$, in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let $g_d(P)$ denote the number of different removal orders we can attain while erasing all points of $P$ this way, and let $g_d(n)$ be the \emph{minimum} of $g_d(P)$ over all $n$-element point sets $P\subset \mathbb{R}^d$. Dumitrescu and Tóth showed that $g_d(n)=(d+1)^{(d+1)^2n}$. We substantially improve their bound, by proving that $g_d(n)= O((d+d\ln{(d)})^{(2+\frac{(d-1)}{\lfloor d\ln{d}\rfloor})n})$. It follows that, for any $ε>0$, there exist sufficiently high dimensional point sets $P\subset \mathbb{R}^d$ with $g_d(P)\leq O(d^{(2+ε)n})$. This almost closes the gap between the upper bound and the best-known lower bound $(d+1)^n$ for large values of $d$. |
| title | The minimum number of peeling sequences of a point set |
| topic | Combinatorics 52C45 |
| url | https://arxiv.org/abs/2312.00244 |