Upper bounds for Heilbronn's triangle problem in higher dimensions
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866913263514550272 |
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| author | Zakharov, Dmitrii |
| author_facet | Zakharov, Dmitrii |
| contents | We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$ contains
- $d+1$ points which span a simplex of volume at most $C_d n^{-\log d+ 6}$,
- $1.1 d$ points whose convex hull has volume at most $C_d n^{-1.1}$,
- $k\ge 4\sqrt{d}$ points which span a $(k-1)$-dimensional simplex of volume at most $C_d n^{-\frac{k-1}{d} - \frac{k^2}{8d^2}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_15715 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Upper bounds for Heilbronn's triangle problem in higher dimensions Zakharov, Dmitrii Combinatorics Metric Geometry We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$ contains - $d+1$ points which span a simplex of volume at most $C_d n^{-\log d+ 6}$, - $1.1 d$ points whose convex hull has volume at most $C_d n^{-1.1}$, - $k\ge 4\sqrt{d}$ points which span a $(k-1)$-dimensional simplex of volume at most $C_d n^{-\frac{k-1}{d} - \frac{k^2}{8d^2}}$. |
| title | Upper bounds for Heilbronn's triangle problem in higher dimensions |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2211.15715 |