Let $n\in\mathbb N$, let $\lambda>0$, and let $\nu$ be a Borel probability measure on $\mathbb R^n$ with finite second moment. For a locally Lipschitz function $h:\mathbb R^n\to\mathbb R$, define its Euclidean local slope by
When $h$ is differentiable at $x$, this local slope agrees with the Euclidean norm of the classical gradient. Assume that $\nu$ satisfies the logarithmic Sobolev inequality with constant $\lambda$ in the following Euclidean local-slope form: for every bounded locally Lipschitz function $g:\mathbb R^n\to\mathbb R$ satisfying
for such densities $f$. Let $W_2$ denote the quadratic Wasserstein distance on Borel probability measures on $\mathbb R^n$ with finite second moments, defined by
where $\Pi(\mu,\nu)$ is the set of Borel couplings of $\mu$ and $\nu$. Then every Borel probability measure $\mu$ on $\mathbb R^n$ such that $\mu\ll\nu$, $\mu$ has finite second moment, and $\operatorname{Ent}_\nu(\mu)<\infty$ satisfies