  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.
