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Given an open, bounded and connected set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Lipschitz boundary and volume $\left|\mathrm{\Omega }\right|$, we prove that the sequence ${\mathcal{F}}_k$ of Dirichlet functionals defined on $H^1\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$, with volume constraints $v^k$ on $m\ge 2$ fixed level-sets, and such that ${\sum }_{i=1}^m{v}_i^k<\left|\mathrm{\Omega }\right|$ for all $k$, $\mathrm{\Gamma }$-converges, as $v^k\to v$ with ${\sum }_{i=1}^m{v}_i^k=\left|\mathrm{\Omega }\right|$, to the squared total variation on $BV\left(V;{\mathbb{R}}^d\right)$, with $v$ as volume constraint on the same level-sets.

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,...,c_m$ in the inequality $\delta P\left(E\right)\ge {\sum }_{k=1}^m{c}_k\alpha {\left(E\right)}^k+o\left(\alpha {\left(E\right)}^m\right)$, valid for each Borel set $E$ with positive and finite area, with $\delta P\left(E\right)$ and $\alpha \left(E\right)$ being, respectively, the $\mathrm{𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡}$ and the $\mathrm{𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦}$ of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $\mathrm{𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠}$ including the lower semicontinuous extension of $\frac{\delta P\left(E\right)}{\alpha {\left(E\right)}^2}$, we describe the...