Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne
Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures
An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential....
Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
We investigate Prékopa-Leindler type inequalities on a Riemannian manifold equipped with a measure with density where the potential and the Ricci curvature satisfy for all , with some . As in our earlier work [], the argument uses optimal mass transport on , but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to...
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