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

In this paper we explicitly estimate the number of points in a subset $A \subset \R^{d}$asafunctionofthemaximumangle$\angle A$ that any three of these points form, provided $\angle A < \theta_d := \arccos(-\frac 1 {d}) \in (\pi/2,\pi)$\mathrm{.}Wealsoshow$\angle A < \theta_d$ensuresthat$A$ 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.