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13
A mass transportation approach to quantitative isoperimetric inequalities
 Invent. Math
, 2010
"... 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 ..."
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Cited by 72 (22 self)
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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 BrenierMcCann Theorem. A sharp quantitative version of the BrunnMinkowski inequality for convex sets is proved as a corollary. 1.
A Selection Principle for the Sharp Quantitative Isoperimetric Inequality
"... Abstract. We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a ..."
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Cited by 34 (4 self)
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Abstract. We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in Rn. Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in R2 in the small asymmetry regime. 1.
THE SHARP SOBOLEV INEQUALITY IN QUANTITATIVE FORM
"... Abstract. A quantitative version of the sharp Sobolev inequality in W 1,p (R n), 1 < p < n, is established with a remainder term involving the distance from extremals. 1. Introduction and ..."
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Cited by 30 (4 self)
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Abstract. A quantitative version of the sharp Sobolev inequality in W 1,p (R n), 1 < p < n, is established with a remainder term involving the distance from extremals. 1. Introduction and
Asymmetric affine Lp Sobolev inequalities
 J. Funct. Anal
"... A new sharp affine Lp Sobolev inequality for functions on Rn is established. This inequality strengthens and implies the previously known affine Lp Sobolev inequality which in turn is stronger than the classical Lp Sobolev inequality. 1. ..."
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Cited by 17 (4 self)
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A new sharp affine Lp Sobolev inequality for functions on Rn is established. This inequality strengthens and implies the previously known affine Lp Sobolev inequality which in turn is stronger than the classical Lp Sobolev inequality. 1.
stability theorems for the anisotropic Sobolev and logSobolev inequalities on functions of bounded variation
 Adv. Math
"... Abstract. Combining rearrangement techniques with Gromov’s proof (via optimal mass transportation) of the 1Sobolev inequality, we prove a sharp quantitative version of the anisotropic Sobolev inequality on BV (Rn). As a corollary of this result, we also deduce a sharp stability estimate for the ani ..."
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Cited by 14 (5 self)
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Abstract. Combining rearrangement techniques with Gromov’s proof (via optimal mass transportation) of the 1Sobolev inequality, we prove a sharp quantitative version of the anisotropic Sobolev inequality on BV (Rn). As a corollary of this result, we also deduce a sharp stability estimate for the anisotropic 1logSobolev inequality.
A note on Cheeger sets
, 2008
"... Starting from the quantitative isoperimetric inequality [21, 17], we prove a sharp quantitative version of the Cheeger inequality. ..."
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Cited by 7 (2 self)
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Starting from the quantitative isoperimetric inequality [21, 17], we prove a sharp quantitative version of the Cheeger inequality.
Bounds on the deficit in the logarithmic sobolev inequality
, 2013
"... Abstract. The deficit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and informationtheoretic distances. 1. ..."
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Cited by 6 (1 self)
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Abstract. The deficit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and informationtheoretic distances. 1.
Uniform PoincaréSobolev and relative isoperimetric inequalities for classes of domains
, 2013
"... ..."
BEST CONSTANTS FOR THE ISOPERIMETRIC INEQUALITY IN QUANTITATIVE FORM
"... Abstract. We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in Rn. Combining these results with a refinement of the selection principle introduced in [11], we describe a method suitable for the determination of the best constants in the quantitative iso ..."
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Abstract. We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in Rn. Combining these results with a refinement of the selection principle introduced in [11], we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesen’s annular symmetrization in a very elementary way, we show that, for n = 2, the abovementioned constants can be explicitly computed through a oneparameter family of convex sets known as ovals. This proves a further extension of a conjecture posed by Hall in [20]. Contents
STABILITY THEOREMS FOR GAGLIARDONIRENBERGSOBOLEV INEQUALITIES: A REDUCTION PRINCIPLE TO THE RADIAL CASE
"... Abstract. In this paper we investigate the quantitative stability for GagliardoNirenbergSobolev inequalities. The main result is a reduction theorem, which states that, to solve the problem of the stability for the GNS inequalities, one can consider only the class of radial decreasing functions. 1 ..."
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Abstract. In this paper we investigate the quantitative stability for GagliardoNirenbergSobolev inequalities. The main result is a reduction theorem, which states that, to solve the problem of the stability for the GNS inequalities, one can consider only the class of radial decreasing functions. 1.