# 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 $\mathbb{R}^n$?

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.

• I don't know about this function, but if you take absolute values instead, then this is called "Manhattan-metric". – klirk Dec 6 '18 at 20:29
• $\frac{1}{N}D(x,y)$ is the average difference in respective components of $x,y$. – AlexanderJ93 Dec 6 '18 at 20:32
• @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 '18 at 20:50
• ^it's linear in $x- y$ so it must satisfy the triangle inequality (with equality). – stochasticboy321 Dec 6 '18 at 20:55
• 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 '18 at 20:57

## 1 Answer

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.