[proofplan] We first establish the Gagliardo--Nirenberg--Sobolev inequality $\|u\|_{L^{n/(n-1)}} \leq C_n\|\nabla u\|_{L^1}$ for $u \in C_c^\infty(\mathbb{R}^n)$ using the fundamental theorem of calculus along each coordinate axis and the Loomis--Whitney inequality. The extension to $BV(U)$ uses mollification: approximate $u \in BV$ by smooth $u_\varepsilon = u * \rho_\varepsilon$, apply the smooth inequality, and pass to the limit using $L^1$ convergence and the [lower semicontinuity of total variation](/theorems/597). [/proofplan] [step:Prove the Gagliardo--Nirenberg--Sobolev inequality for smooth compactly supported functions] For $u \in C_c^\infty(\mathbb{R}^n)$ and each $i \in \{1, \ldots, n\}$, the fundamental theorem of calculus gives: \begin{align*} |u(x)| \leq \int_{-\infty}^\infty |\partial_i u(x_1, \ldots, t, \ldots, x_n)| \, d\mathcal{L}^1(t). \end{align*} Therefore: \begin{align*} |u(x)|^{n/(n-1)} \leq \prod_{i=1}^n \left(\int_{-\infty}^\infty |\partial_i u| \, d\mathcal{L}^1(t_i)\right)^{1/(n-1)}. \end{align*} Integrating over $x \in \mathbb{R}^n$ and applying the Loomis--Whitney inequality (an iterated application of H\"older's inequality in the $n-1$ complementary variables): \begin{align*} \|u\|_{L^{n/(n-1)}}^{n/(n-1)} \leq \prod_{i=1}^n \|\partial_i u\|_{L^1}^{1/(n-1)} \leq \|\nabla u\|_{L^1}^{n/(n-1)}, \end{align*} where the last inequality uses AM-GM: $\prod_i \|\partial_i u\|_{L^1}^{1/(n-1)} \leq (n^{-1}\sum_i \|\partial_i u\|_{L^1})^{n/(n-1)} \leq \|\nabla u\|_{L^1}^{n/(n-1)}$. [/step] [step:Extend to $BV(U)$ by mollification and lower semicontinuity] For $u \in BV(U)$, let $u_\varepsilon = u * \rho_\varepsilon$ be the mollification. Then $u_\varepsilon \in C^\infty$ and $\|\nabla u_\varepsilon\|_{L^1(V)} \leq |Du|(U)$ for any $V \Subset U$ with $\operatorname{dist}(V, \partial U) > \varepsilon$ (this bound follows from the convolution estimate and the [Dual Characterisation](/theorems/591)). Applying the smooth inequality to $u_\varepsilon$ on $V$: \begin{align*} \|u_\varepsilon\|_{L^{n/(n-1)}(V)} \leq C_n\,|Du|(U). \end{align*} As $\varepsilon \to 0$: $u_\varepsilon \to u$ in $L^1(V)$, and a subsequence converges a.e. Fatou's lemma gives $\|u\|_{L^{n/(n-1)}(V)} \leq C_n\,|Du|(U)$. Letting $V \nearrow U$ via monotone convergence: \begin{align*} \|u\|_{L^{n/(n-1)}(U)} \leq C\,\|u\|_{BV(U)}. \end{align*} [/step]