Pentagon Minimization without Computation
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914957125222400 |
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| author | Mackey, John Subercaseaux, Bernardo |
| author_facet | Mackey, John Subercaseaux, Bernardo |
| contents | Erdős and Guy initiated a line of research studying $μ_k(n)$, the minimum number of convex $k$-gons one can obtain by placing $n$ points in the plane without any three of them being collinear. Asymptotically, the limits $c_k := \lim_{n\to \infty} μ_k(n)/\binom{n}{k}$ exist for all $k$, and are strictly positive due to the Erdős-Szekeres theorem. This article focuses on the case $k=5$, where $c_5$ was known to be between $0.0608516$ and $0.0625$ (Goaoc et al., 2018; Subercaseaux et al., 2023). The lower bound was obtained through the Flag Algebra method of Razborov using semi-definite programming. In this article we prove a more modest lower bound of $\frac{5\sqrt{5}-11}{4} \approx 0.04508$ without any computation; we exploit``planar-point equations'' that count, in different ways, the number of convex pentagons (or other geometric objects) in a point placement. To derive our lower bound we combine such equations by viewing them from a statistical perspective, which we believe can be fruitful for other related problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17098 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pentagon Minimization without Computation Mackey, John Subercaseaux, Bernardo Combinatorics Computational Geometry Discrete Mathematics Erdős and Guy initiated a line of research studying $μ_k(n)$, the minimum number of convex $k$-gons one can obtain by placing $n$ points in the plane without any three of them being collinear. Asymptotically, the limits $c_k := \lim_{n\to \infty} μ_k(n)/\binom{n}{k}$ exist for all $k$, and are strictly positive due to the Erdős-Szekeres theorem. This article focuses on the case $k=5$, where $c_5$ was known to be between $0.0608516$ and $0.0625$ (Goaoc et al., 2018; Subercaseaux et al., 2023). The lower bound was obtained through the Flag Algebra method of Razborov using semi-definite programming. In this article we prove a more modest lower bound of $\frac{5\sqrt{5}-11}{4} \approx 0.04508$ without any computation; we exploit``planar-point equations'' that count, in different ways, the number of convex pentagons (or other geometric objects) in a point placement. To derive our lower bound we combine such equations by viewing them from a statistical perspective, which we believe can be fruitful for other related problems. |
| title | Pentagon Minimization without Computation |
| topic | Combinatorics Computational Geometry Discrete Mathematics |
| url | https://arxiv.org/abs/2409.17098 |