A mass transportation approach to quantitative isoperimetric inequalities

By Alessio Figalli, Francesco Maggi, Aldo Pratelli. 2010

448 citations recorded by the source corpus.

Subfields: Applied Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/915137078a997bd3aef42c6dbdfb3fd1836508a0

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Citations in the corpus

Quantitative stability for sumsets in $R^n$ (2014, methodology): “Furthermore, if we restrict A and B to the class of convex sets, a quantitative stability estimate for the Brunn-Minkowski inequality, with the sharp power law dependence on the deficit, was proved in [6, 7].”
On the isoperimetric deficit in Gauss space (2011, background): “THE ISOPERIMETRIC DEFICIT IN GAUSS SPACE 135 inequality was established (see also [19], and [15] for the anisotropic case).”
A Selection Principle for the Sharp Quantitative Isoperimetric Inequality (2010, methodology): “In fact, in [15,19] it is proved that, for some c > 0, inf Q c/4n and inf Q c/n6, respectively.”
Optimal transportation, topology and uniqueness (2010, background): “The first problem has become classical in the calculus of variations; it has deep connections to analysis [30,51,73,74,98], geometry [29,63,70,71,79,83,95,104], dynamics [7,10,59,82] and nonlinear partial differential equations [18,20,21,36,42, 72,102], as well as applications in physics [38,75,97],”
On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique (2014, background): “2) is the recent paper [11], where stability issues are addressed as well.”
Remainder terms in the fractional Sobolev inequality (2012, background): “ (and isoperimetric inequalities) has again attracted a lot of attention in the last years. The recent works use techniques from symmetrization (see, e.g., [8,13]), optimal transportation (see, e.g., [12]), and fast diﬀusion (see, e.g., [10,11,17]); see also [7] for a recent application of remainder”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, unknown): “Integrating inequality (8) on M+ yields (9) V ol(M+) ≤ 1 n ∫ M+ ∆u dnx = 1 n ∫ ∂M+ Du(x) · n̂M+(x)dHn−1(x).”
Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, methodology): “All the proofs in [18, 61, 29] employ the following so-called “Poincaré-type trace inequality” proved by Figalli–Maggi–Pratelli in [18].”
Sharp gradient stability for the Sobolev inequality (2020, background): “Introduction Motivated by important applications to problems in the calculus of variations and evolution PDEs, in recent years there has been a growing interest around the understanding of quantitative stability for functional/geometric inequalities, see for instance [3, 2, 8, 27, 28, 21, 9, 22, 29,”
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation (2011, background): “Moreover, other recent proofs for stability based on optimal transportation [17] or symmetrization techniques [19, 20, 13, 18] did not produce (at least up to now) any results in this situation.”
A User’s Guide to Optimal Transport (2013, methodology): “The use of Brenier’s map instead of Knothe’s one makes the difference when studying the quantitative version of the isoperimetric problem: Figalli, Maggi and Pratelli in [38], using tools coming from optimal transport, proved the sharp quantitative isoperimetric inequality in R endowed with any norm”
Quantitative logarithmic Sobolev inequalities and stability estimates (2014, result): “Examples of such results are the recent quantitative stability estimates for Sobolev [12, 19], Brunn-Minkowski [17, 16], and isoperimetric inequalities [20, 18, 15, 23].”
On the Shape of Compact Hypersurfaces with Almost‐Constant Mean Curvature (2015, background): “n; g/m: By the quantitative isoperimetric inequality [15, 17], one finds xm 2 RnC1 such that jm.”