Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical distribution on the $x_i$'s, we'd just be looking for the cheapest way to assign a mass of $1/n$ from $f$ to each of the points. In other words we'd solve the problem $$\min_{f_1,\dots,f_n} \sum_{i=1}^n\int_D f_i(x)\|x-x_i\|dx$$ subject to the constraints that $f_i\geq0$ for all $i$ and that $\int_D f_i(x) dx = 1/n$ for all $i$, as well as the constraint that $f = f_1 + \cdots + f_n$. Each $f_i$ represents the mass assigned to point $x_i$.

My question is: is there a name for the quantity that I would get if I change the sum in the objective function to a maximum over $i$, and remove the constraint that $\int_D f_i(x) dx = 1/n$? In other words, is there a name for the optimal cost to the problem $$\min_{f_1,\dots,f_n} \max_{i}\int_D f_i(x)\|x-x_i\|dx$$ subject to the constraints that $f_i(x)\geq0$ and $f = f_1 + \cdots + f_n$? This seems like a natural quantity but I cannot find any references dealing with it.