  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
[24]. The stability for the Gaussian logarithmic Sobolev inequality was
obtained as a byproduct of the stability for the Sobolev inequality. Here we
give a new, direct, alternative proof. We also discuss improved versions of
interpolation inequalities based on the carr\'e du champ method.
