# Modulus of continuity of parameterizing Wasserstein

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?

$$\newcommand\De\Delta\newcommand\de\delta$$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.