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

Abstract

We show that there exists an absolute positive constant $b(\ge \frac{1}{48})$ 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 $\Vert \cdot \Vert$ 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 $\Vert \cdot \Vert$.