  We prove a sharp quantitative version for the stability of the Sobolev
inequality with explicit constants. Moreover, the constants have the correct
behavior in the limit of large dimensions, which allows us to deduce an optimal
quantitative stability estimate for the Gaussian log-Sobolev inequality with an
explicit dimension-free constant. Our proofs rely on several ingredients such
as competing symmetries, a flow based on continuous Steiner symmetrization that
interpolates continuously between a function and its symmetric decreasing
rearrangement, and refined estimates on the Sobolev functional in the
neighborhood of the optimal Aubin--Talenti functions.
