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

TL;DR

This work addresses how many distinct directions and distances a $d$-dimensional point set in $\R^d$ can determine. It combines a high-dimensional extension of Ungar's theorem with a bichromatic, generalized-segment framework and central projections, leveraging a DS&Wigderson bound to prove a linear-in-$n$ bound: there exists $b \,\ge\, \frac{1}{48}$ such that a $d$-dimensional set not contained in a hyperplane determines at least $b d n$ lines with pairwise distinct directions. Furthermore, for typical $d$-norms, the paper shows the existence of $(b d - o(1)) n$ distinct distances for large $n$, aligning with a broader conjecture on dimensional scaling. The results illuminate a linear-growth regime in high dimensions and contrast with the Euclidean case, suggesting Theta$(n d)$ behavior for typical norms and highlighting connections to Dirac-type problems in discrete geometry.

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 $\|\cdot \|$.

Figures (8)

• Figure 1: Generalized segments determined by red and blue points. The dotted portion on the right is not part of the generalized segment determined by the corresponding red point and blue point.
• Figure 2: Pairs of (co-planar) convergent generalized segments. The dotted portions are not part of the corresponding generalized segments.
• Figure 3: The indexing of the points of $P$ in Lemma \ref{['lemma:2d']}.
• Figure 4: Part 1 of Lemma \ref{['lemma:2d']}. On the left the case where $p_{1}$ and $p_{n}$ (here $n=9$) are separated by $\ell$. On the right the case where $p_{1}$ and $p_{n}$ are not separated by $\ell$.
• Figure 5: Part 2 of Lemma \ref{['lemma:2d']}. On the left the case where $p_{i}$ and $p_{i+1}$ (here $p_{7}$ and $p_{8}$) are separated by $\ell$. On the right the case where $p_{i}$ and $p_{i+1}$ (here $p_{5}$ and $p_{6}$) are not separated by $\ell$.

Theorems & Definitions (13)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Lemma 2.1
• Theorem 2.2: Theorem 1.9 in DSW14
• Corollary 2.3
• Claim 2.4