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It is known that the sublevel sets of the relative entropy are tight when the reference measure is finite, and in fact are also compact in the topology of setwise convergence (which is stronger than t …

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\mathb …

Yes, it turns out you can do better than the triangle inequality in this case. See section 3.1 of "Faster Wasserstein Distance Estimation with the Sinkhorn Divergence".

Yes, there are various results available in more general settings. The typical route would be to combine an upper bound on the expected distance between the law and the empirical measure (like Theorem …

In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\epsilon$-approximate leng …

Let $(X,d)$ be a complete separable metric space, and let $(\mathcal{P}_2 (X), W_2)$ be the space of probability measures on $X$ with finite second moments, equipped with the 2-Wasserstein distance. I …

Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$. Does it hold …