Distinct Directions and Distinct Distances in ℝ^d

2025/08/12 by Noga Alon, Alon, Noga, Rom Pinchasi +1
Computer Science · Mathematics · #52C10 #52C30 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematical Approximation and Integration #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2508.08870

openalex publication_date 2025/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that there exists an absolute positive constant b (≥ (1)/(48)) so that any set of n points in 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 ‖⋅‖ so that every set of n>n₀(d) points that is d-dimensional determines at least (bd-o(1))n distinct distances with respect to ‖⋅ ‖.

Citations

• Unit and distinct distances in typical norms
• Improved rank bounds for design matrices and a new proof of Kelly's theorem
• Rank Bounds for Design Matrices with Applications to Combinatorial\n Geometry and Locally Correctable Codes
• COLLINEARITY PROPERTIES OF SETS OF POINTS

Related

• Unit and distinct distances in typical norms
• Distinct distances between a collinear set and an arbitrary set of\n points
• On sets of directions determined by subsets of \Bbb R^d
• A remark on sets with few distances in ℝ^d
• Erdős Distance Problem in ℝ^d