There's a [paper that discusses the behaviour of distance metrics in high dimensional spaces](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.23.7409&rep=rep1&type=pdf). 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 $f \in (0..1)$.