Sum of angles in $\mathbb{R}^n$ Given three vectors $v_1,v_2$ and $v_3$ in $\mathbb{R}^n$ with the standard scalar product the follwing is true
$$\angle(v_1,v_2)+\angle(v_2,v_3)\geq \angle(v_1,v_3).$$
It tried to substitute $\angle(v_1,v_2) = cos^{-1}\frac{v_1 \cdot v_2}{\Vert v_1 \Vert \Vert v_2 \Vert}$ but I could not show the resulting inequality. What is the name of the inequality and do you know reference that one can cite in an article?

You can reduce the problem to $\mathbb{R}^3$, and there wlog v2=(0,0,1)
then one gets an easy to prove inequality if one writes everything in polar coordinates.
