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**Inserted:** 11 apr 2011

**Last Updated:** 13 sep 2016

**Journal:** Proceedings of the Bourbaki Seminar

**Year:** 2011

**Notes:**

in French

**Abstract:**

The classical isoperimetric inequality prescribes the maximal volume of a body in $R^n$ under a perimeter constraint. Since the optimum is given by the ball $B$, it gives
$P(E)\geq n

E

^{1-1/n}

B

^{1/n}$ for any $E$ in $\mathbb R^n$. Its anisotropic version concerns the $K-$perimeter $P_K$, defined from a convex body $K$ in $\mathbb R^n$, and it reads $P_K(E)\geq n

E

^{1-1/n}

K

^{1/n}$; the optimum is realized by $K$. A quantitative version of these inequalities means estimating the difference $P(E)-n

E

^{1-1/n}

B

^{1/n}$ in terms of ``how much $E$ is different from $B$''. The optimal quantitative version of the classical inequality has been proved in 2008 by Fusco, Maggi and Pratelli through some symmetrization methods, peculiar to the isotropic case. The works that I'll present have allowed, thanks to the application of the Brenier transport, to get the same result in the anisotropic case.

**Keywords:**
BV functions, trace Sobolev inequality, Brenier Transport, Knothe transport

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