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 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: | 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 \ | :<math> Per(E) \geq d \omega_n^{1/n}|E|^{1-1/n}. </math> | ||
Here <math> \omega_n </math> is the volume of the unit ball in <math> \mathbb{R}^n </math>. The idea of the proof is to construct a map T called Knothe transport and use it between the two densities: <math> \mu=\mathcal{L}_B </math>, the inequality will follow from some symmetries and consideration on the Jacobian determinant of this map. | |||
==The Knothe's transport== | |||
For this part I will follow the first chapter in <ref> A. Figalli, F. Glaudo An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows </ref>. In some sense, it can be seen as a multidimension generalization of monotone rearrangement. | |||
Take any two measures <math> \mu,\nu</math> and define | |||
:<math> F(x)= \int_{-\infty}^x d\mu(t), \text{ } G(y)= \int_{-\infty}^x d\nu(t) </math>, | |||
This maps may not be well defined, since at some points the measures may have a delta. For the purpose of this exposition we will assume that those functions are well defined, for the precise definition and convention to include the mass of the deltas in the integral we refer again to the first chapter of<ref> A. Figalli, F. Glaudo An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows </ref>. | |||
Revision as of 00:38, 12 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:
Here is the volume of the unit ball in . The idea of the proof is to construct a map T called Knothe transport and use it between the two densities: , the inequality will follow from some symmetries and consideration on the Jacobian determinant of this map.
The Knothe's transport
For this part I will follow the first chapter in [4]. In some sense, it can be seen as a multidimension generalization of monotone rearrangement. Take any two measures and define
- ,
This maps may not be well defined, since at some points the measures may have a delta. For the purpose of this exposition we will assume that those functions are well defined, for the precise definition and convention to include the mass of the deltas in the integral we refer again to the first chapter of[5].
- ↑ 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)
- ↑ A. Figalli, F. Glaudo An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows
- ↑ A. Figalli, F. Glaudo An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows