Reference request for Isoperimetric Inequality using Sobolev inequality

Question

I want to prove Isoperimetric inequality from Sobolev inequality. I have seen a post in stack , but I want to know if some proper reference exists which I didn't find yet which proves Isoperimetric inequality from Sobolev inequality without using Calculas of Variations. Any help would be very helpful.

Answer 1

There is a proof on 5.6.2 of Evans and Gariepy's Measure Theory and Fine Properties of Functions (revised edition) using Sobolev's inequality for BV functions: There exists $C>0$ such that for all $f \in BV(\mathbb{R}^{n})$ we have $$ \|f\|_{L^{\frac{n}{n-1}}(\mathbb{R}^{n})} \leq C \|Df\|(\mathbb{R}^{n}).$$ This is Theorem 5.10. Now, Theorem 5.11 states that the same constant satisfies $$\mathcal{L}(E)^{\frac{n-1}{n}} \leq C \|\partial E\|(\mathbb{R}^{n}),$$ where $E$ is a bounded set of finite perimeter in $\mathbb{R}^{n}$. To prove it, just apply Theorem 5.10 with $f=\chi_{E}$.