Exponentially sized pointsets with angles less than 61 degrees
Miroslav Marinov

TL;DR
This paper establishes exponential bounds on the size of point sets in high-dimensional space where all angles are less than a specified threshold slightly above 60 degrees, using geometric and combinatorial methods.
Contribution
It provides tight exponential bounds for the maximum size of such point sets, refining previous approaches and connecting geometric packing problems with hypergraph intersection problems.
Findings
Point set size grows as (1+Theta(c))^d for small c
Upper bounds relate to sphere cap packing problems
Lower bounds involve hypergraph intersection problems
Abstract
We prove that any set of points in , any three of which form an angle less than , has size for sufficiently small . The proof is based on a refinement of an approach by Erd\H{o}s and F\"{u}redi. The lower bound is relying on a problem about large hypegraphs with small edge intersections, while the upper bound is tightly connected to the problem of packing disjoint caps on a sphere.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputational Geometry and Mesh Generation · Point processes and geometric inequalities · Optimization and Packing Problems
