Let $R$ be the real numbers. I want to consider metrics $d: R^n \times R^n \rightarrow R$ such that there are functions $h: R \rightarrow R$ and $\delta: R \times R \rightarrow R$ with $$ d(x,y) = h \left(\sum_{i=1}^n \delta(x_i,y_i) \right).$$
This applies, for example, to the Euclidean distance, or more generally, to the metric induced by the $p$-norm with $$h(z) = z^{1/p}$$ and $$\delta(a,b) = |a-b|^p$$ resulting in $$d(x,y) = \left(\sum_{i=1}^n |x_i-y_i|^p \right)^{1/p}. $$
In particular, I need this sum structure over the coordinates $i$ to build the metric. My questions:
- Does there exist a name for this kind of metric?
- Are there important examples of metrics possessing this structure beyond the $p$-norm example? I mean, otherwise it would not make much sense to consider this generalization.
