Heilbronn's triangle problem in three dimensions
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909878605316096 |
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| author | Maldague, Dominique Wang, Hong Zakharov, Dmitrii |
| author_facet | Maldague, Dominique Wang, Hong Zakharov, Dmitrii |
| contents | We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's triangle problem. This estimate is a consequence of the following result about configurations of point-line pairs in $\mathbb R^3$: for $n \ge 2$ let $p_1, \ldots,p_n \in [0,1]^3$ be a collection of points and let $\ell_i$ be a line through $p_i$ for every $i$ such that $d(p_i, \ell_j) \ge δ$ for all $i\neq j$. Then we have $n \lesssim δ^{-3+γ}$ for some absolute constant $γ>0$. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26644 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heilbronn's triangle problem in three dimensions Maldague, Dominique Wang, Hong Zakharov, Dmitrii Combinatorics Classical Analysis and ODEs Metric Geometry We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's triangle problem. This estimate is a consequence of the following result about configurations of point-line pairs in $\mathbb R^3$: for $n \ge 2$ let $p_1, \ldots,p_n \in [0,1]^3$ be a collection of points and let $\ell_i$ be a line through $p_i$ for every $i$ such that $d(p_i, \ell_j) \ge δ$ for all $i\neq j$. Then we have $n \lesssim δ^{-3+γ}$ for some absolute constant $γ>0$. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author. |
| title | Heilbronn's triangle problem in three dimensions |
| topic | Combinatorics Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2510.26644 |