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The standard functional isoperimetric inequality is for an integer $n\ge 1$,
$$
\Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le c(n)\Vert \nabla u\Vert_{L^1(\mathbb R^n)}, \quad c(n)=\frac{(\vert\mathbb B^n_2\vert_n)^{\frac{n-1}{n}}}{
\vert\mathbb S^{n-1}_2\vert_{n-1}
},
\tag{$\flat$}$$
where $\...
