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

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.

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