Given a Minkowski (or Finsler) space $(V,F)$, I am wondering how to define the angle between two vectors $w$ and $v$. I first thought it must be as $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(v,v)}}.$$ Indeed I fixed the inner product $g_w$ and then defined the angle using it. I am also suspected that one can define the angle as $$\cos\theta(w,v)=\frac{g_w(w,v)}{F(v)F(w)}.$$

Any idea?

P.S. by $g_w$ I mean the second fundamental form which is defined as $$g_w(v,u)=\frac{1}{2}\frac{\partial^2 F^2}{\partial t \partial s}(w+su+tv)|_{s=t=0}.$$