# Transportation-cost inequality for pushforward measure

Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies Talagrand transportation-cost inequality, namely

There exists a constant $c_\mu > 0$ such that $$W(\nu,\mu) \le \sqrt{2c_\mu H(\nu\|\mu)},$$ for every other probability measure $\nu$ on $X$.

Here, $H$ denotes relative entropy (KL divergence) and $W$ denotes the Wasserstein $2$-distance on the space of probability measures on $X$ (with finite 2nd moment).

# Question

Does the pushforward $\varphi_\#\mu$ of $\mu$ under $\varphi$ satisfy such an inequality ? My guess is that it does, with constant $c_{\varphi_\#\mu} \le L c_\mu$.

The result you want (but of course with $c_{\varphi_\#\mu} \le L^2 c_\mu$ rather than $c_{\varphi_\#\mu} \le L c_\mu$) is Lemma 2.1 in the paper at https://arxiv.org/pdf/math/0410172
• That's awesome. Thanks. Also thanks for catching the $L \implies L^2$ thinko :) Sep 3 '18 at 18:13