I want to define a "distance" between two subsets $A, B$ of a normed space $(V, \|\cdot\|)$ both with (at most) $n$ elements. A straightforward way for me to do this would be to define

$$ d(A, B) := \min_ {\pi\in S_n}\\ \sum_{i=i}^n \|a_i - b_{\pi(i)}\| $$

where $A = (a_1, \ldots, a_n)$ and $B$ likewise with $b_i$. (The image to have in mind: find that matching of points in $A$ and $B$ that produces the smallest distance sum.)

My question is: Has this already been done, i. e. is this a well-known concept?