ETD

• continuous symmetrization
• isoperimetric inequality
• optimal transport
• quantitative stability
• selection principle
• sharp dimensional estimates
• Sobolev inequality
• spectral gap

This thesis provides an overview of known results concerning the quantitative Euclidean isoperimetric inequality and related functional inequalities. The primary focus is the investigation of the asymptotic behavior of the optimal constant as the dimension n tends to infinity.
We present an ongoing attempt to determine the precise asymptotic growth of this constant, following a roadmap proposed by Prof. Figalli:
- Local Expansion: Establishing the bound for sets satisfying weak graphicality conditions over the ball.
- Local to Global: Implementing a reduction procedure to substitute generic sets with the aforementioned graphical ones.
This framework is inspired by the approach successfully applied to the quantitative Sobolev inequality for p=2 by Dolbeault, Esteban, Figalli, Frank and Loss, which is analyzed here in detail.