# Dependency of the Wasserstein metric on its parameters

Let the population on some region $$\Omega\subset\mathbb R^d$$ be modeled by a density function $$\rho:\Omega\to (0,+\infty)$$. Provided $$n\ge 1$$ food trucks labeled by their capacity $$p_1,\ldots, p_n\in (0,\infty)$$, where

$$\sum_{i=1}^n p_i ~~=~~ \int_{\Omega} \rho(x)dx ~~=~~ 1.$$

We pursue the "best distribution" of the trucks. Namely, if the trucks are located at $$y_1,\ldots, y_n\in\Omega$$, then the distance in average between the inhabitants and the source (food) is quantified by the Wasserstein distance $$W_2$$, i.e.

$$F(Y)~~:=~~W_2^2\big(\mu_Y,\nu\big),$$

where the discrete measure $$\mu_Y:=\sum_{i=1}^np_i\delta_{y_i}(dx)$$ stands for the distribution of the trucks, with $$Y:=(y_1,\ldots, y_n)$$, and $$\nu$$ is the probability measure with density $$\rho$$, i.e. $$\nu(dx):=\rho(x)dx$$. My concern is to investigate the regularity of $$F$$ on $$\Omega^n$$, i.e. $$\nabla F$$, and the minimization problem $$\inf_{Y\in\Omega^n}F(Y)$$. Any ideas or comments are highly appreciated!

Some thoughts: If $$\Omega$$ is assumed to be compact, then the minimizer for $$F$$ must exist. Also, it follows by definition that $$Y\mapsto F(Y)$$ is Lipschitz, as

$$\big|W_2(\mu_Y,\nu)-W_2(\mu_Y',\nu)\big|~~\le~~W_2(\mu_Y,\mu_Y')~~\le~~\sum_{i=1}^n|y_i-y_i'|,$$

where the first inequality follow from the triangle inequality and the second one from the dual formulation of $$W_2$$.

• Some questions: What is the ground metric on $\mathbb{R}^d$ for the Wasserstein distance? If $F(Y)$ is the square of the Wasserstein 2 distance, how do your thoughts show Lipschitz continuity (your inequality is without the square)? – Steve May 1 '19 at 8:49
• @Steve Indeed, I am always assuming the case of compact $\Omega$. Then for any $Y\in\Omega^n$, $W(\mu_Y,\nu)=\inf_{\lambda}\big(\int_{\Omega}\lambda d\mu_Y-\int_{\Omega}\lambda d\nu \big) \le 2(C+1)$, where the inf is taken overall $1-$Lipschitz functions $\lambda:\Omega\to\mathbb R$, and $C$ in the second inequality denotes a upper bound of $\Omega$, i.e. $\sup_{x\in\Omega} |x|\le C$. – Neymar May 1 '19 at 11:17
• @Neymar: Then your notations are confusing. Your definition above $\inf_\lambda$ is the 1- Wasserstein distance $W_1(\mu,\nu)$ (also known as 'earth mover' or 'bounded-Lipschitz). The notation $W_2$ should rather stand for the quadratic Wasserstein distance with cost function $c(x,y)=|x-y^2$. So write $W_1$ everywhere. – leo monsaingeon May 9 '19 at 8:03