There's a paper that discusses the behaviour of distance metrics in high dimensional spaces. They take on the $L_k$ norm and propose the manhattan $L_1$ norm as the most effective in high dimensional spaces for clustering purposes. They also introduce a fractional norm $L_f$ similar to the $L_k$ norm but with $f \in (0..1)$.

The paper On the surprising behavior of distance metrics in high dimensional space discusses the behaviour of distance metrics in high dimensional spaces. 

They take on the $L_k$ norm and propose the manhattan $L_1$ norm as the most effective in high dimensional spaces for clustering purposes. They also introduce a fractional norm $L_f$ similar to the $L_k$ norm but with $f \in (0..1)$.

In short, they show that for high dimensional spaces using the euclidean norm as a default is probably not a good idea; we have usually little intuition in such spaces, and the exponential blowup due to the number of dimensions is hard to take into account with the euclidean distance.