Complete logarithmic Sobolev inequalities via Ricci curvature bounded below

Modified logarithmic Sobolev inequalities in null curvature

We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that Φ is a symmetric convex function on R satisfying (1 + ε)Φ(x) 6 xΦ(x) 6 (2 − ε)Φ(x) for x > 0 large enough and with ε ∈]0, 1/2]. We prove that the probability measure on R μΦ(dx) = e /ZΦdx satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant A,B,D > 0 such that for al...

Metric measure spaces with Riemannian Ricci curvature bounded from below

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, i...

Modified logarithmic Sobolev inequalities and transportation inequalities

Inequalities on the Ricci curvature

We improve Chen-Ricci inequalities for a Lagrangian submanifold Mn of dimension n (n 2) in a 2n -dimensional complex space form M̃2n(4c) of constant holomorphic sectional curvature 4c with a semi-symmetric metric connection and a Legendrian submanifold Mn in a Sasakian space form M̃2n+1(c) of constant φ -sectional curvature c with a semi-symmetric metric connection, respectively.

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for d...