lp norm
Families of Distance Function $l_p$ norm The most common one is $l_2$ norm (Euclidean distance):
$$||x - z||_2 = \sqrt{\sum_{i=1}^{m}(x_i - z_i)^2}$$
Notes: sometime 2 is dropped.
For $p \geq 1$, the $l_p$ distance:
$$||x - z||_p = (\sum_{i=1}^{m}(x_i - z_i)^p)^{1/p}$$
Special case:
$l_1$ distance: $$||x - z||_1 = \sum_{i=1}^{m}|x_i - z_i|$$
$l_\infty$ distance:
$$||x - z||_1 = max_i |x_i - z_i|$$
Metric space Let $X$ be the space that data lie.……