[proofplan] The isoperimetric inequality reduces to the Gagliardo--Nirenberg--Sobolev inequality for $BV$ applied to the characteristic function $\mathbb{1}_E$: since $\|\mathbb{1}_E\|_{L^{n/(n-1)}} = \mathcal{L}^n(E)^{(n-1)/n}$ and $|D\mathbb{1}_E|(\mathbb{R}^n) = \operatorname{Per}(E; \mathbb{R}^n)$, the [BV Embedding](/theorems/594) directly yields the inequality. The sharp constant and equality case follow from the P\'olya--Szeg\H{o} inequality and symmetric decreasing rearrangement. [/proofplan] [step:Apply the $BV$ embedding to $\mathbb{1}_E$ to obtain the isoperimetric inequality] Let $E$ be a set of finite perimeter with $\mathcal{L}^n(E) < \infty$. Then $\mathbb{1}_E \in BV(\mathbb{R}^n)$ with $\|\mathbb{1}_E\|_{L^1} = \mathcal{L}^n(E)$ and $|D\mathbb{1}_E|(\mathbb{R}^n) = \operatorname{Per}(E; \mathbb{R}^n)$. Compute: \begin{align*} \|\mathbb{1}_E\|_{L^{n/(n-1)}} = \left(\int_{\mathbb{R}^n} |\mathbb{1}_E|^{n/(n-1)} \, d\mathcal{L}^n\right)^{(n-1)/n} = \mathcal{L}^n(E)^{(n-1)/n}. \end{align*} The Gagliardo--Nirenberg--Sobolev inequality for $BV$ (the [BV Embedding](/theorems/594) extended to $\mathbb{R}^n$ for compactly supported functions) gives: \begin{align*} \mathcal{L}^n(E)^{(n-1)/n} = \|\mathbb{1}_E\|_{L^{n/(n-1)}} \leq C_n\,|D\mathbb{1}_E|(\mathbb{R}^n) = C_n\,\operatorname{Per}(E; \mathbb{R}^n). \end{align*} [/step] [step:Identify the sharp constant and the equality case] The sharp constant is $C_n = 1/(n\omega_n^{1/n})$ where $\omega_n = \mathcal{L}^n(B(0,1))$. For the equality case: the symmetric decreasing rearrangement of $\mathbb{1}_E$ is $\mathbb{1}_{B^*}$ where $B^*$ is a ball with $\mathcal{L}^n(B^*) = \mathcal{L}^n(E)$. The P\'olya--Szeg\H{o} inequality gives $\operatorname{Per}(B^*) \leq \operatorname{Per}(E)$, with equality if and only if $E$ is a ball up to $\mathcal{L}^n$-null sets. One verifies that balls achieve equality: for $B = B(0, r)$, $\mathcal{L}^n(B) = \omega_n r^n$ and $\operatorname{Per}(B) = n\omega_n r^{n-1}$, so: \begin{align*} \frac{\mathcal{L}^n(B)^{(n-1)/n}}{\operatorname{Per}(B)} = \frac{(\omega_n r^n)^{(n-1)/n}}{n\omega_n r^{n-1}} = \frac{\omega_n^{(n-1)/n}}{n\omega_n} = \frac{1}{n\omega_n^{1/n}} = C_n. \end{align*} [/step]