Isometries in metric $L_n$ Let's define distance between the points in 2-d space: $$d = \sqrt[n]{(x_1 - x_2) ^ n + (y_1 - y_2) ^ n},\quad n > 2$$
The isometries (linear transformations that preserves the length of vectors) in this space seem confusing to me. How to define the angle between vectors? 

The isometries of this space are the symmetries of the square, dihedral group with 8 elements. This is because the unit ball for the metric has those symmetries and no others; an isometry must map it onto itself. 
Generally, one doesn't define an angle between vectors in a normed space. If you really want to, take a look at the semi-inner product 
$$ [u,v] = \frac{\sum {u_iv_i|v_i|^{n-2}}}{\left(\sum |v_i|^{n}\right)^{(n-2)/n}}$$
which has a bit of useful properties such as $[u,u]=\|u\|^2$, $|[u,v]|\le \|u\|\|v\|$ and linearity in $u$. 
But it's non-symmetric, so $\cos^{-1}\left(\frac{[u,v]}{\|u\|\|v\|}\right)$ and $\cos^{-1}\left(\frac{[v,u]}{\|v\|\|u\|}\right)$ give you two different angle values. 
