On the cardinality of sets in ${\bf R}^d$ obeying a slightly obtuse angle bound

TL;DR

This paper estimates the maximum number of points in a subset of Euclidean space with a restricted maximum angle between triples, showing such sets form convex polytopes, addressing questions related to Erdős and Fejes Tóth.

Contribution

It provides explicit bounds on the size of point sets in ^d with angle constraints and characterizes their geometric structure as convex polytopes.

Findings

Derived explicit bounds on point set cardinality based on angle restrictions.

Proved that sets with angles less than a specific threshold are convex polytope vertices.

Connected geometric angle constraints to classical problems in discrete geometry.

Abstract

In this paper we explicitly estimate the number of points in a subset $A \subset \R^{d}$ as a function of the maximum angle $\angle A$ that any three of these points form, provided $\angle A < \theta_d := \arccos(-\frac 1 {d}) \in (\pi/2,\pi)$. We also show $\angle A < \theta_d$ ensures that $A$ coincides with the vertex set of a convex polytope. This study is motivated by a question of Paul Erd\H{o}s and indirectly by a conjecture of L\'aszl\'o Fejes T\'oth.