Modulus of continuity of parameterizing Wasserstein

Question

Let $x_1,\dots,x_n\in X$ some Polish space $X$ and let $\Delta$ be the probability simplex in $\mathbb{R}^n$. Consider the map sending every $(w_1,\dots,w_n)\in\Delta$ to the finitely supported measure $\sum_{k=1}^n w_k\delta_{x_k}$. This map is clearly continuous with respect to the Wasserstein distance, but is it also Lipschitz?

Answer 1

$\newcommand\De\Delta\newcommand\de\delta$Yes, this map is Lipschitz. Indeed, the map is \begin{equation*} \De\ni w=(w_1,\dots,w_n)\mapsto\mu_w:=\sum_{k=1}^n w_k\de_{x_k}. \tag{1} \end{equation*} Let $d$ denote the metric on $X$, and then let \begin{equation*} D:=\max_{i,j\in[n]}d(x_i,x_j), \end{equation*} the diameter of the set $\{x_1,\dots,x_n\}$, where $[n]:=\{1,\dots,n\}$.

Take any $v=(v_1,\dots,v_n)$ and $w=(w_1,\dots,w_n)$ in $\De$ and let \begin{equation*} h:=\max_{j\in[n]}|v_j-w_j|. \end{equation*} Consider the following $(n-1)$-step transportation plan of transporting the probability measure $\mu_v$ to $\mu_w$ (or vice versa).

For the first step of the plan, transport $\mu_v$ to $\mu_{v^{(1)}}$, where \begin{equation*} v^{(1)}:=\Big(w_1,v_2+\frac{v_1-w_1}{n-1},\dots,v_n+\frac{v_n-w_n}{n-1}\Big), \end{equation*} assuming without loss of generality that $v_1\ge w_1$. Then (i) the first coordinate of $v^{(1)}$ is the same as that of $w$, (ii) $v^{(1)}\in\De$, (iii) \begin{equation*} \max_{2\le j\le n}|v^{(1)}_j-w_j|\le\max_{2\le j\le n}|v_j-w_j|+\frac{v_1-w_1}{n-1}\le h+\frac h{n-1}=\frac n{n-1}\,h, \end{equation*} and (iv) \begin{equation*} W(\mu_v,\mu_{v^{(1)}})\le\sum_{j=2}^n \frac{v_1-w_1}{n-1}\,d(x_1,x_j) \le(v_1-w_1)D\le hD, \end{equation*} where $W$ denotes the Wasserstein distance.

In the remaining $n-2$ steps of the plan, similarly and consecutively equalizing the remaining $n-1$ coordinates of the initially given vectors $v=(v_1,\dots,v_n)$ and $w=(w_1,\dots,w_n)$, we see that \begin{equation*} W(\mu_v,\mu_w)\le hD+\frac n{n-1}\,hD+\frac n{n-2}\,hD+\cdots+\frac n1\,hD \le Lh, \end{equation*} where $L:=n(1+\ln n)D$. So, the map (1) is Lipschitz.