Name for a pseudometric: $d(x,y) = \inf_{\sigma} \lVert x- y\circ\sigma\rVert_1$ I would like to know if the following notion of distance* has been considered in the literature, and if so under which name(s):
For $x,y\in\mathbb{R}^n$, let
$$
d(x,y) =  \inf_{\sigma\in\mathcal{S}_n} \lVert x- y\circ\sigma\rVert_1 = \inf_{\sigma\in\mathcal{S}_n} \sum_{i=1}^n \lvert x_i - y_{\sigma(i)}\rvert
$$
where $\mathcal{S}_n$ is the symmetric group on $\{1,\dots,n\}$, i.e. the group of all permutations. Equivalently, $d(x,y)$ is the $\ell_1$ distance between the vectors $x,y$ after sorting their coordinates.
(The asterisk above is because this does not define a distance, but instead a pseudometric: it doesn't satisfy the axiom of identity of indiscernibles.) 
Is there a standard name for $d$? "Sorted $\ell_1$ distance" does not seem to bear many results...
PS: I am actually mostly interested as $d$ as a pseudometric on $\Delta(\{1,\dots,n\})$, i.e. over the set of vectors with non-negative entries summing to one. But I figure the general case is as relevant.
 A: I don't know of an attested name, but I can suggest a reasonable name.
Consider these two metric spaces:


*

*Consider first $X=(\mathbb R^n,\ell^1)$, the Euclidean space with the $\ell^1$ norm.
The permutation group $S_n$ acts on $\mathbb R^n$ by permuting the components.
Let $Q=X/S_n$ be the quotient of the space $X$ by this action, obtained from $X$ by identifying different points on the same orbit.
The quotient inherits a metric from $X$ in the standard way.

*Let $Y=(\mathbb R^n,d)$ be the Euclidean space with your semimetric.
A semimetric gives rise to an equivalence relation given by
$$
x\sim y
\iff
d(x,y)=0.
$$
Once you take the quotient $P=Y/{\sim}$, you are left with a metric space.
The semimetric $d$ gives rise to a metric on $P$, and in fact I would even say that the metric on $P$ is $d$.
The $d$-distance is independent of choices of representatives of an equivalence class, so everything is well defined.
The point is that the spaces $X$ and $Y$ can be naturally identified, and this identification is an isometry.
The quotient by the action of the permutation group is precisely the quotient by the relation $\sim$.
I have not seen this semimetric before, because of this structure, I would call it the permutation quotient $\ell^1$ metric (as a metric on the quotient $Q=P$ or a semimetric on $\mathbb R^n$ by metonymy), or the $\ell^1$ norm modulo permutations.
