The five gradients inequality for non quadratic costs (Q2687982)

The paper contains a proof of the so-called ``five gradients inequality'' of optimal transport, valid for cost functions of the form \(c(x,y)=h(x-y)\) where \(h\in C^1(\mathbb{R}^d)\) is strictly convex and radially symmetric. The result can thus be applied to all the \(p\)-Wasserstein distances for \(p>1\), extending the previously known case of \(p=2\). The five gradients inequality is telling us that, given two probability densities \(\rho,g\in W^{1,1}(\Omega)\) of a convex domain \(\Omega\) and a radially symmetric convex function \(H\in C^1(\mathbb{R}^d\setminus\{0\})\), then \[ \int_{\Omega} \left(\nabla\rho\cdot\nabla H(\nabla \varphi)+\nabla g\cdot\nabla H(\nabla \psi)\right)dx\ge 0 \] where \((\varphi,\psi)\) are Kantorovich potentials for the problem between \(\rho\) and \(g\) and cost \(h\). This inequality can be used to derive some \(\mathrm{BV}\) and Sobolev estimates for solutions of the \(\mathrm{JKO}\) scheme.