Is there a name for the function $D(x,y) = \sum\limits_{i = 1}^n (x_i - y_i)$, where $x$ and $y$ are vectors in $\mathbb{R}^n$?

Question

I am studying real analysis and encountered this function

$$D(x,y) = \sum\limits_{i = 1}^n (x_i - y_i)$$

where $x,y$ are $n$-dimensional vectors living in $\mathbb{R}^n$.

It is easy to show that is it not a distance.

I wonder if there is a name for this function.

Answer 1

This isn't a full answer but I think it does give insight

$|D(x,y)|$ as defined above is the shortest distance (Manhattan Metric) of $y$ from the plane $P=\{x': \sum_{i=1}^n x'_i = \sum_{i=1}^n x_i \}$. The sign of $D(x,y)$ captures to which side of $P$ is $y$ on.