Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,x_n\in E$ and $\rho_1,\ldots,\rho_n\ge0$ with $\sum_{i=1}^n\rho_i=1$ such that $$\mu:=\sum_{i=1}^n\rho_i\delta_{x_i},\tag0$$ where $\delta$ denotes the Dirac kernel on $E$, "captures $\nu$ well".
The latter may clearly be described by a suitable metric on the space of probability measures on $\mathcal B(E)$; they are choosing the $L^p$-Wasserstein metric $\mathcal W_p$ for some $p\ge1$.
However, now they seem to claim (on p. 4) that if $\rho_1=\cdots=\rho_n=n^{-1}$ and $x_1,\ldots,x_n$ are chosen such that $\mu$ $$\mathcal W_p(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\:\nu)}\left(\int\pi({\rm d}(x,y))d(x,y)^p\right)^{\frac1p},$$ where $\Pi(\mu,\nu)$ denotes the set of couplings of $\mu$ and $\nu$, is minimized (with respect to the choice of $x_1,\ldots,x_n$), then the sampling pattern corresponding to $\mu$ has "blue noise".
However, I don't really understand what the latter is supposed to mean and I cannot find any formal verification that the minimizer above actually has this property. Assume $E=\mathbb R^d$ for some $d\in\mathbb N$. Then, usually, "blue noise" should mean that the Fourier transform $$\hat\mu(x)=\sum_{i=1}^n\rho_ie^{-{\rm i}2\pi\langle x,\:x_i\rangle}\;\;\;\text{for }x\in\mathbb R^d$$ has "low enegery for small frequencies". And this should mean that $|\hat\mu(x)|$ is small whenever $|x|$ is small. But how can we prove that this is the case for the minimizer above?
