Semester paper on isoperimetric inequalities.
In this semester paper we go through the work of N. Fusco, F. Maggi and A. Pratelli in [The sharp quantitative isoperimetric inequality] as shown by F. Maggi in the survey [Some methods for studying stability in isoperimetric type problems], explaining the details in this approach towards proving a sharp stability result for the classical isoperimetric inequality in \( \mathbb{R}^n \). After this, we study a generalized version of the problem: the anisotropic isoperimetric inequality, and show stability as done by A. Figalli, F. Maggi and A. Pratelli in [A mass transportation approach to quantitative isoperimetric inequalities].