In this paper, we present recent stability results with explicit and dimensionally sharp constants and optimal norms for the Sobolev inequality and for the Gaussian logarithmic Sobolev inequality obtained by the authors in [22]. The stability for the Gaussian logarithmic Sobolev inequality is obtained as a byproduct of the stability for the Sobolev inequality. Here we also give a new, direct, alternative proof of this result. We also discuss improved versions of interpolation inequalities based on the carré du champ method.
