Isoperimetric inequality and OMT: Difference between revisions
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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. <ref>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)</ref>. This proof is based on an idea by Knothe <ref> Herbert Knothe. "Contributions to the theory of convex bodies.." Michigan Math. J. 4 (1) 39 - 52, 1957 <ref> | |||
==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. <ref>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)</ref>. This proof is based on an idea by Knothe <ref>Herbert Knothe. "Contributions to the theory of convex bodies.." Michigan Math. J. 4 (1) 39 - 52, 1957</ref>. The classic isoperimetric inequality in <math> \mathbb{R}^n </math> 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 <math> E </math> has a larger perimeter than the ball <math> B </math> with the same volume. I will present this proof following the exposition given in chapter two in <ref> F. Santambrogio. Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs and Modeling (2015) </ref>. The usually way to state this is the following: | |||
:<math> Per(E) \geq d \omega_d^{1/d}|E|^{1-1/d}. </math> | |||
Revision as of 23:40, 11 February 2022
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 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 has a larger perimeter than the ball 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:
- ↑ 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)