Definition:General Euclidean Metric

Let $$\reals^n$$ be an $n$-dimensional real vector space.

Let $$x = \left({x_1, x_2, \ldots, x_n}\right) \in \reals^n$$ and $$y = \left({y_1, y_2, \ldots, y_n}\right) \in \reals^n$$.

Consider the Euclidean metric $$d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{\frac 1 2}$$ on $$\reals^n$$.

The generalized Euclidean metric is defined as follows:

$$ $$ $$

The Generalized Euclidean Metric is a Metric.

Note that $$d_2 \left({x, y}\right)$$ is the usual Euclidean metric.