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arXiv · 2508.08870

Distinct Directions and Distinct Distances in $\mathbb{R}^d$

Noga Alon·Rom Pinchasi
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Abstract

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 $\|\cdot \|$.

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BibTeXRIS

Noga Alon, Rom Pinchasi. 2025-11-07. Distinct Directions and Distinct Distances in $\mathbb{R}^d$. https://arxiv.org/abs/2508.08870

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