Upper bound of Wasserstein distance given by subvariables of codim 1

Question

recently I am considering the upper-bound of Wasserstain distance. Say we have random vectors $X,Y$ of dimension $n$, and let $\tilde{X}_i (\tilde{Y}_i,$ resp.) be the $(n-1)$-dim random vector of $X (Y $, resp.) discarding the $i$-th component. For example, $n=3$ and $X=(X_1,X_2,X_3)$, then $\tilde{X}_2=(X_1,X_3)$.

My question is, can we formulate an inequality of the form $W_p(X,Y) \leq \sum \limits_{i=1}^na_i W_p(\tilde{X}_i,\tilde{Y}_i)$? I know that we can formulate similar inequality by using $1$-dim marginals refer here, hence I believe such inequality would hold for $(n-1)$-subvariables.

Answer 1

$\newcommand\tX{\tilde X}\newcommand\tY{\tilde Y}\newcommand\D{\overset D=}$The answer is no. E.g., suppose that $n=2$, so that $\tX_1,\tX_2,\tY_1,\tY_2$ can be identified with $X_2,X_1,Y_2,Y_1$, respectively, and the inequality in question becomes $$W_p(X,Y)\le a_1W_p(X_2,Y_2)+a_2W_p(X_1,Y_1).\tag{1}$$ Suppose also $X_1\D Y_1$, $X_2\D Y_2$, but $X\not\D Y$, where $\D$ denotes the equality in distribution. Then the right-hand side of (1) is $0$ but the left-hand side is $>0$. So, (1) fails to hold.