Suppose I have $\vec x = (x_1, \ldots, x_n)$ and $\vec y = (y_1, \ldots, y_n)$ and a sequence of metrics $d_i$ for $i \in \{ 1, \ldots, n\}$ that are used for the $i$th component.
Consider a vector of distances, $\vec d (\vec x, \vec y)$ where the $i$th component of $\vec d (\vec x, \vec y)$ is given by $d_i(x_i,y_i)$.
Now to further aggregate the distances. Some aggregation function $\alpha : [0,\infty)^n \mapsto \mathbb{R}$ could be considered. More narrowly there may be choices of aggregation functions that composed on our distance vector would give a distance function on the vectors $\vec x$ and $\vec y$.
$\alpha (d_1(x_1, y_1), \ldots, d_n(x_n, y_n))$
Example Take $d_i$ to be the taxicab distance, and $\alpha$ to be the $p$th root of the sum of powers of $p$ of its inputs. $$||\vec x - \vec y||_p = \sqrt[p]{\sum_{i}^n |x_i-y_i|^p}$$ is a metric.
Question If $\alpha$ is a norm, will a norm on a vector of metrics always produce a metric between the vectors $\vec x$ and $\vec y$ assuming they are vectors from $\mathbb{R}^n$?