Uniform log-Sobolev inequalities for mean field particles beyond flat-convexity

Abstract: In the nice recent work [48], S. Wang established uniform log-Sobolev inequalities for mean field particles when the energy is flat convex. In this note we comment how to extend his proof to some semi-convex energies provided the curvature lower-bound is not too negative. It is not clear that this could be obtained simply by applying a posteriori a perturbation argument to Wang's result. Rather, we follow his proof and, at steps where the convexity is used, we notice that the uniform conditional or local functional inequalities assumed give some room to allow for a bit of concavity. In particular, this allows to recover other previous results on non-convex systems at high temperature or weak coupling, and to consider situations which mix flat-convexity and weak coupling.