The role played by p in the process of minimizing V_p is to control the importance of the size of the fluctuations away from the mean value. The larger p is, the more large fluctuations are penalized compared to small ones. The greatest advantage of the Euclidean norm, p=2, is that it allows for analytic solutions and it guarantees that V_p will only have a single minimum. When p\neq 2 there can be multiple minima in V_p, making it difficult to ensure that a particular minimum found will be the global minimum instead of a local one.
