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.

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    I don't know about this function, but if you take absolute values instead, then this is called "Manhattan-metric". – klirk Dec 6 at 20:29
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    $\frac{1}{N}D(x,y)$ is the average difference in respective components of $x,y$. – AlexanderJ93 Dec 6 at 20:32
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    @Mike It's also antisymmetric and not positive in general. Haven't checked if it verifies triangle inequality, but I'd bet it doesn't. – AlexanderJ93 Dec 6 at 20:50
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    ^it's linear in $x- y$ so it must satisfy the triangle inequality (with equality). – stochasticboy321 Dec 6 at 20:55
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    This is just a comment, but I'd view it as the dot product $(\mathbf x-\mathbf y)\cdot \langle1,\ldots,1\rangle$. – Mark S. Dec 6 at 20:57

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.

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