During the seminar, we will introduce a stronger version of the quantitative isoperimetric inequality, originally developed by Fusco and Julin. Building on that, we will arrive to the analogous inequality for a capillarity problem using the so-called selection principle, based on the regularity theory for the perimeter functional. We will also highlight the difficulties that arise when we apply Fusco and Julin’s method to our situation. This result was obtained in collaboration with Giulio Pascale and Marco Pozzetta.