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There is a bound, the coupling $(\xi, a \xi)$ leads to the upper bound \begin{align*} W_p(\xi, a \xi)^p &\leq \mathbb{E}[|\xi - a\xi|^p] = |(1-a)|^p \mathbb{E}[|\xi|^p]\\ \Rightarrow ~~~~ W_p(\xi, a\x …

Perhaps another simple argument that $\mathcal{X}$ is indeed equal to $\mathcal{W}_2(\mathbb{R}^d$): Starting from the fact that there is a bimeasurable bijection $b : \mathbb{R}^d \rightarrow \mathbb …

I hope I did not misunderstand the question, but it seems $\varphi(x) > - \infty$ holds as follows if $(x, y) \in \Gamma$: For any $(x_i, y_i) \in \Gamma$, $i=1, \dots, n$, we see that \begin{align} …

This is just a partial answer regarding $S$ being injective. The generalisation of your argument is given by Hornik (Theorem 5 and the definition of discriminatory functions above Theorem 5)

I think I have an answer for the case p = 1, K = 2. I write "I think" because my computation does not coincide with the example values for $N=4$ posted earlier by OP in a comment, but I really cannot …

As already mentioned in the comments, this is really a standard result, see, e.g., Lemma 3.1. in Kallenberg's book.