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

TL;DR

This paper establishes lower bounds on the number of distinct directions and distances determined by large, d-dimensional point sets in Euclidean space, revealing fundamental geometric properties.

Contribution

It proves the existence of a universal constant ensuring many distinct directions and distances in d-dimensional point sets, extending understanding of geometric configurations.

Findings

At least bd n lines with distinct directions in any d-dimensional point set.

Existence of d-dimensional norms with many distinct distances for large point sets.

Quantitative bounds on the number of distinct directions and distances.

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 $\|\cdot \|$.