[dnautics]

maxnorm in 3-dimensions is not euclidean. It is however, a metric space (which is why they call them "balls"). Note that the following formula for vectors x and y over R is a valid metric (obeys triangle inequality) for _all_ n in _all_ dimensions, with n=2 being the usual euclidean metric:

    d(x, y) = |sum_i((x_i - y_i)^n)|^(1/n)

The maxnorm is the limit as n -> infinity, and is also a proper metric space. (the collection of these spaces are called Lp spaces)

Of course in one dimension, all of these Lp metrics are identical.

https://en.wikipedia.org/wiki/Lp_space