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It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...

Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$]. For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, we consider ...

$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth. Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...

I'm reading a monograph that considers the following problem: $$\inf_{z(t) \in C^1} \int_0^1 c\bigg(\frac{dz(t)}{dt}\bigg) dt\\ z(0) = x, z(1) = y$$ Here $c$ is a convex function, $z(t)$ are paths ...