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

Abstract

In this paper we explicitly estimate the number of points in a subset $A\subset {\mathbb{R}}^d$ 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 Erdős and indirectly by a conjecture of László Fejes Tóth.

arXiv commentv4 will be published in the SIAM Journal on Discrete Mathematics