The isoperimetric inequality (Q6610705)

This paper is a short but very pleasant-to-read review on the isoperimetric and Sobolev inequalities. After stating the isoperimetric inequality, it introduces the Sobolev inequality in \(\mathbb{R}^N\) and sketches how this latter implies the former. It then describes at large two ways to prove the Sobolev inequality: Gromov's argument via Knothe's rearrangement, and the ABP method. Finally, it introduces the topic of Sobolev inequalities on hypersurfaces.