I'm trying to prove the following statement:
Prove that every set of more than $8$ points in the three-dimensional space determines at least one obtuse angle.
I'm aware that the generalization of this question for $\mathbb{R}^d$ was proved in 1962 by Danzer and Grunbaum. But I'm interested in a prove just for the case $d=3$ using convex polytopes or similar arguments to the case $d=2$. Anyone know about such proof? Help will be appreciated.