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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.

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  • $\begingroup$ You're using the word "sample" incorrectly. You should say $x_1,\ldots,x_n$ is a sample from $f$, not that $x_1,\ldots,x_n$ are samples from $f$. ${}\qquad{}$ $\endgroup$ – Michael Hardy Jul 27 '15 at 15:59
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This is the more from decision theory, which could also be regarded as part of statistics if you regard the procedure of making a statistical decision as sort of de-randomization from a categorical viewpoint like Jared&Sturtz. Here the loss function is chosen to be $ f_i(x)\|x-x_i\|$ (like weighting over a compact domain $D$) and the risk is actually calcualted using uniform distribution on $D$. I think it can be formalized less twisted using a suitable model under the name of minimax decision problem. Now it is not a solvable minimax problem.

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