For c-uniformly convex potentials V, the log-Sobolev and Talagrand deficits control L1 distance to extremizers with exponent 1/19, improved to the optimal 1/2 for radial densities.
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Quantitative stability for Bakry--\'Emery log-Sobolev and Talagrand inequalities
For c-uniformly convex potentials V, the log-Sobolev and Talagrand deficits control L1 distance to extremizers with exponent 1/19, improved to the optimal 1/2 for radial densities.