The sharp constant in the Sobolev inequality and the set of optimizers are known. It is also known that functions whose Sobolev quotient is almost minimial are close to minimizers. We are interested in a quantitative version of the last statement and present a bound that not only measures this closeness in the optimal topology and with the optimal exponent, but also has explicit constants. These constants have the optimal 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 proof relies on several ingredients:
• a discrete flow based on competing symmetries;
• a continuous rearrangement flow;
• refined estimates in the neighborhood of the optimal Aubin-Talenti functions.
The talk is based on joint work with Dolbeault, Esteban, Figalli and Loss.