Isoperimetric inequality and OMT

The classic isoperimetric inequality

A very interesting application of optimal transport is a proof of the isoperimetric inequality. The first proof with an OMT argument is due to Gromov and the main tool is the Knothe's map. ^[1]. This proof is based on an idea by Knothe ^[2]. The classic isoperimetric inequality in $\mathbb {R} ^{n}$ states that the round ball has the minimal (n-1)-dimensional volume of the boundary among all the domains with a given fixed volume. This is equivalent to say that every set $E$ has a larger perimeter than the ball $B$ with the same volume. I will present this proof following the exposition given in chapter two in ^[3]. The usually way to state this is the following:

Per(E)\geq d\omega _{d}^{1/d}|E|^{1-1/d}.

↑ V.D. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, with an appendix by M. Gromov, Lecture notes in Mathematics, vol. 1200 (Springer, Berlin, 1986)
↑ Herbert Knothe. "Contributions to the theory of convex bodies.." Michigan Math. J. 4 (1) 39 - 52, 1957
↑ F. Santambrogio. Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs and Modeling (2015)

[1] V.D. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, with an appendix by M. Gromov, Lecture notes in Mathematics, vol. 1200 (Springer, Berlin, 1986)

[2] Herbert Knothe. "Contributions to the theory of convex bodies.." Michigan Math. J. 4 (1) 39 - 52, 1957

[3] F. Santambrogio. Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs and Modeling (2015)

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Isoperimetric inequality and OMT

The classic isoperimetric inequality

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