Distinct Directions and Distinct Distances in $\mathbb{R}^d$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909893734170624 |
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| author | Alon, Noga Pinchasi, Rom |
| author_facet | Alon, Noga Pinchasi, Rom |
| contents | We show that there exists an absolute positive constant $b (\geq \frac{1}{48})$ so that any set of $n$ points in $\mathbb{R}^d$ that is $d$-dimensional determines at least $bdn$ lines with pairwise distinct directions. As a consequence we prove that there are $d$-dimensional real norms $\|\cdot\|$ so that every set of $n>n_0(d)$ points that is $d$-dimensional determines at least $(bd-o(1))n$ distinct distances with respect to $\|\cdot \|$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08870 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinct Directions and Distinct Distances in $\mathbb{R}^d$ Alon, Noga Pinchasi, Rom Combinatorics 52C10, 52C30 We show that there exists an absolute positive constant $b (\geq \frac{1}{48})$ so that any set of $n$ points in $\mathbb{R}^d$ that is $d$-dimensional determines at least $bdn$ lines with pairwise distinct directions. As a consequence we prove that there are $d$-dimensional real norms $\|\cdot\|$ so that every set of $n>n_0(d)$ points that is $d$-dimensional determines at least $(bd-o(1))n$ distinct distances with respect to $\|\cdot \|$. |
| title | Distinct Directions and Distinct Distances in $\mathbb{R}^d$ |
| topic | Combinatorics 52C10, 52C30 |
| url | https://arxiv.org/abs/2508.08870 |