# Questions tagged [optimal-transportation]

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### What is the role of of continuity in this proof of Kantorovich duality?

I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation. Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability measures ...
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### Why is $\Xi \equiv 0$ if $E=C_{0}(X \times Y)$?

I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation. In page 28, the author said that Exercise 1.11. Let us try to extend this proof to the non-compact ...
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### Nontrivial lower-bound for $\mathbb P(f(X) \le \alpha \|\nabla f(X)\|)$

Let $f:\mathbb R^d \to \mathbb R$ be a differentiable function whose gradient is $L$-Lipschitz (w.r.t euclidean norm), for some fixed $L \in [0,\infty)$. Let $X=(X_1,\ldots,X_d)$ be a concentrated ...
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### Estimate of Wasserstein distance and flow of vector fields under particular assumptions

Let $\mu$ be a compactly supported absolutely continuous probability measure. Let $v,u$ be Lipschitz vector fields. For a vector field $w$ recall that $\Phi_t^w$ denotes its flow. A classical estimate ...
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### Orthogonality in Wasserstein tangent space for discrete measures with equal mass

Let say I have $N$ discrete probability measures $(\mu_1,...,\mu_N)$ where each of them has $n$ points in $\mathbb{R}^2$ of equal mass. Let $P(\mathcal{X})$ be the space of these probability measures ...
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### Reweighting probability measures by convex potentials, and contraction in transport distance

Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by $$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$ i.e. each $\mu_y$ is a ...
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### Semiconcavity estimate for the squared distance on a compact Riemannian manifold

I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...