For two probability measures $\mu,\nu$ on $\mathbb{R}^n$, let $$W_p(\mu,\nu)^p=\inf_{\pi}\int\|x-y\|^p_2\ \pi(dx\ dy)$$ denote the $p^{th}$ Wasserstein distance between $\mu, \nu$, where the infimum is taken with respect to all possible couplings of $\mu,\nu$. Consider probability measures $\mu_1,\ldots, \mu_n$ and $\nu_1,\ldots, \nu_n$ on $\mathbb{R}$. For $p\ge 1$, is it true that $$W_p(\mu_1\otimes\cdots\otimes\mu_n, \nu_1\otimes \cdots\otimes\nu_n)\le \left(\sum_{i=1}^n W^2_p(\mu_i,\nu_i)\right)^{1/2} \ \ \ ?$$

For two probability measures $\mu,\nu$ on $\mathbb{R}^n$, let $$W_p(\mu,\nu)^p=\inf_{\pi}\int\|x-y\|^p_2\ \pi(dx\ dy)$$ denote the $p^\text{th}$ Wasserstein distance between $\mu, \nu$, where the infimum is taken with respect to all possible couplings of $\mu,\nu$. Consider probability measures $\mu_1,\ldots, \mu_n$ and $\nu_1,\ldots, \nu_n$ on $\mathbb{R}$. For $p\ge 1$, is it true that $$W_p(\mu_1\otimes\cdots\otimes\mu_n, \nu_1\otimes \cdots\otimes\nu_n)\le \left(\sum_{i=1}^n W^2_p(\mu_i,\nu_i)\right)^{1/2} \ \ \ \text{?}$$

For two probability measures $\mu,\nu$ on $\mathbb{R}^n$, let $$W_p(\mu,\nu)=\inf_{\pi}\int\|x-y\|^p_2\ \pi(dx\ dy)$$ denote the $p^{th}$ Wasserstein distance between $\mu, \nu$, where the infimum is taken with respect to all possible couplings of $\mu,\nu$. Consider probability measures $\mu_1,\ldots, \mu_n$ and $\nu_1,\ldots, \nu_n$ on $\mathbb{R}$. For $p\ge 1$, is it true that $$W_p(\mu_1\otimes\cdots\otimes\mu_n, \nu_1\otimes \cdots\otimes\nu_n)\le \left(\sum_{i=1}^n W^2_p(\mu_i,\nu_i)\right)^{1/2} \ \ \ ?$$

For two probability measures $\mu,\nu$ on $\mathbb{R}^n$, let $$W_p(\mu,\nu)^p=\inf_{\pi}\int\|x-y\|^p_2\ \pi(dx\ dy)$$ denote the $p^{th}$ Wasserstein distance between $\mu, \nu$, where the infimum is taken with respect to all possible couplings of $\mu,\nu$. Consider probability measures $\mu_1,\ldots, \mu_n$ and $\nu_1,\ldots, \nu_n$ on $\mathbb{R}$. For $p\ge 1$, is it true that $$W_p(\mu_1\otimes\cdots\otimes\mu_n, \nu_1\otimes \cdots\otimes\nu_n)\le \left(\sum_{i=1}^n W^2_p(\mu_i,\nu_i)\right)^{1/2} \ \ \ ?$$