A mass transportation approach to quantitative isoperimetric inequalities
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In this paper, a sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy.Abstract:
A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary.read more
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Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory
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A User’s Guide to Optimal Transport
Luigi Ambrosio,Nicola Gigli +1 more
TL;DR: In this paper, the authors provide a quick and reasonably account of the classical theory of optimal mass transportation and its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.
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Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies
TL;DR: In this paper, a sharp quantitative isoperimetric inequality for nonlocal s-perimeters, uniform with respect to s bounded away from 0, was obtained for balls of small volume with a competition between perimeter and nonlocal potentials.
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A Selection Principle for the Sharp Quantitative Isoperimetric Inequality
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Minimality via Second Variation for a Nonlocal Isoperimetric Problem
TL;DR: In this paper, the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts is discussed.
References
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TL;DR: In this paper, the metric side of optimal transportation is considered from a differential point of view on optimal transportation, and the Kantorovich duality of the optimal transportation problem is investigated.
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FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs)
TL;DR: By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp.