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2017 | OriginalPaper | Chapter

Stability for Borell-Brascamp-Lieb Inequalities

Authors : Andrea Rossi, Paolo Salani

Published in: Geometric Aspects of Functional Analysis

Publisher: Springer International Publishing

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Abstract

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be L 1-close to be p-concave and to coincide up to homotheties of their graphs.

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Appendix
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Literature
1.
go back to reference S. Artstein, B Klartag, V. Milman, The Santaló point of a function, and a functional form of the Santaló inequality. Mathematika 51 (1–2), 33–48 (2004) S. Artstein, B Klartag, V. Milman, The Santaló point of a function, and a functional form of the Santaló inequality. Mathematika 51 (1–2), 33–48 (2004)
3.
5.
go back to reference H.J. Brascamp, E.H. Lieb, Some inequalities for Gaussian measures and the long-range order of one-dimensional plasma, in Functional Integration and Its Applications, ed. by A.M. Arthurs (Clarendon Press, Oxford, 1975), pp. 1–14 H.J. Brascamp, E.H. Lieb, Some inequalities for Gaussian measures and the long-range order of one-dimensional plasma, in Functional Integration and Its Applications, ed. by A.M. Arthurs (Clarendon Press, Oxford, 1975), pp. 1–14
6.
go back to reference H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 (4), 366–389 (1976)CrossRefMATH H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 (4), 366–389 (1976)CrossRefMATH
7.
go back to reference D. Bucur, I. Fragalà, Lower bounds for the Prékopa-Leindler deficit by some distances modulo translations. J. Convex Anal. 21 (1), 289–305 (2014)MathSciNetMATH D. Bucur, I. Fragalà, Lower bounds for the Prékopa-Leindler deficit by some distances modulo translations. J. Convex Anal. 21 (1), 289–305 (2014)MathSciNetMATH
9.
go back to reference M. Christ, Near equality in the two-dimensional Brunn-Minkowski inequality. Preprint 2012, arXiv:1206.1965v2 M. Christ, Near equality in the two-dimensional Brunn-Minkowski inequality. Preprint 2012, arXiv:1206.1965v2
10.
go back to reference M. Christ, Near equality in the Brunn-Minkowski inequality. Preprint 2012, arXiv:1207.5062v1 M. Christ, Near equality in the Brunn-Minkowski inequality. Preprint 2012, arXiv:1207.5062v1
11.
go back to reference A. Dinghas, Uber eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus. Math. Z. 68, 111–125 (1957)MathSciNetCrossRefMATH A. Dinghas, Uber eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus. Math. Z. 68, 111–125 (1957)MathSciNetCrossRefMATH
12.
go back to reference V.I. Diskant, Stability of the solution of a Minkowski equation. Sibirsk. Mat. Ž. 14, 669–673, 696 (1973) [Russian] V.I. Diskant, Stability of the solution of a Minkowski equation. Sibirsk. Mat. Ž. 14, 669–673, 696 (1973) [Russian]
13.
go back to reference S. Dubuc, Critères de convexité et inégalités intégrales. Ann. Inst. Fourier 27 (1), x, 135–165 (1977) [French. English summary] S. Dubuc, Critères de convexité et inégalités intégrales. Ann. Inst. Fourier 27 (1), x, 135–165 (1977) [French. English summary]
14.
go back to reference R. Eldan, B. Klartag, Dimensionality and the stability of the Brunn-Minkowski inequality. Ann. Sc. Norm. Super. Pisa Cl. Sci. XIII (5), 975–1007 (2014) R. Eldan, B. Klartag, Dimensionality and the stability of the Brunn-Minkowski inequality. Ann. Sc. Norm. Super. Pisa Cl. Sci. XIII (5), 975–1007 (2014)
15.
go back to reference A. Figalli, D. Jerison, Quantitative stability for the Brunn-Minkowski inequality. Preprint 2014, arXiv:1502.06513v1 A. Figalli, D. Jerison, Quantitative stability for the Brunn-Minkowski inequality. Preprint 2014, arXiv:1502.06513v1
16.
go back to reference A. Figalli, F. Maggi, A. Pratelli, A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. Henri Poincare 26, 2511–2519 (2009)MathSciNetCrossRefMATH A. Figalli, F. Maggi, A. Pratelli, A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. Henri Poincare 26, 2511–2519 (2009)MathSciNetCrossRefMATH
17.
go back to reference A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to quantitative isoperimetric inequality. Invent. Math. 182 (1), 167–211 (2010)MathSciNetCrossRefMATH A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to quantitative isoperimetric inequality. Invent. Math. 182 (1), 167–211 (2010)MathSciNetCrossRefMATH
19.
go back to reference D. Ghilli, P. Salani, Quantitative Borell-Brascamp-Lieb inequalities for power concave functions. Preprint (2015) D. Ghilli, P. Salani, Quantitative Borell-Brascamp-Lieb inequalities for power concave functions. Preprint (2015)
21.
go back to reference G. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934)MATH G. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934)MATH
22.
go back to reference R. Henstock, A.M. Macbeath, On the measure of sum sets, I. The theorems of Brunn, Minkowski and Lusternik. Proc. Lond. Math. Soc. 3 (3), 182–194 (1953) R. Henstock, A.M. Macbeath, On the measure of sum sets, I. The theorems of Brunn, Minkowski and Lusternik. Proc. Lond. Math. Soc. 3 (3), 182–194 (1953)
23.
go back to reference B. Klartag, Marginals of geometric inequalities, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1910 (Springer, Berlin, 2007), pp. 133–166 B. Klartag, Marginals of geometric inequalities, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1910 (Springer, Berlin, 2007), pp. 133–166
24.
go back to reference L. Leindler, On a certain converse of Hölder’s inequality, II. Acta Sci. Math. 33 (3–4), 217–223 (1972)MATH L. Leindler, On a certain converse of Hölder’s inequality, II. Acta Sci. Math. 33 (3–4), 217–223 (1972)MATH
25.
go back to reference A. Prékopa, Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. 32, 301–316 (1971)MathSciNetMATH A. Prékopa, Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. 32, 301–316 (1971)MathSciNetMATH
26.
go back to reference R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1993) R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1993)
27.
go back to reference A. Segal, Remark on stability of Brunn-Minkowski and isoperimetric inequalities for convex bodies, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 2050 (Springer, Heidelberg, 2012), pp. 381–391 A. Segal, Remark on stability of Brunn-Minkowski and isoperimetric inequalities for convex bodies, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 2050 (Springer, Heidelberg, 2012), pp. 381–391
Metadata
Title
Stability for Borell-Brascamp-Lieb Inequalities
Authors
Andrea Rossi
Paolo Salani
Copyright Year
2017
DOI
https://doi.org/10.1007/978-3-319-45282-1_22

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