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Stability for the Brunn-Minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry

Published online by Cambridge University Press:  20 November 2018

Eric Carlen  and
Francesco Maggi
Eric Carlen
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854-8019, USA e-mail: carlen@math.rutgers.edu
Francesco Maggi
Affiliation:
Department of Mathematics, University of Texas, Austin TX 78712 e-mail: maggi@math.utexas.edu
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Abstract

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We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.

Keywords

49N99Brunn-Minkowski inequalityRiesz rearrangementGaussian ConcentrationGates-Penrose-Lebowitz energy

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 5 , 01 October 2017 , pp. 1036 - 1063
Copyright
Copyright © Canadian Mathematical Society 2017

References

[BBJ14] Brancolini, A., Barchiesi, M., and Julin, V., Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality. Ann. Probab., to appear, http://cvgmt.sns.it/paper/2516/ Google Scholar
[Bur96] Burchard, A., Cases of equality in the Riesz rearrangement inequality. Ann. of Math. (2) 143(1996), no. 3, 499–527.http://dx.doi.org/10.2307/2118534 Google Scholar
[CCE+09] Carlen, E. A., Carvalho, M. C., Esposito, R., Lebowitz, J. L., and Marra, R., Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional. Nonlinearity 22(2009), no. 12, 2919–2952.http://dx.doi.org/10.1088/0951-7715/22/12/007 Google Scholar
[CFMP11] Cianchi, A., Fusco, N., Maggi, F., and Pratelli, A., On the isoperimetric deficit in Gauss space. Amer. J. Math. 133(2011), no. 1, 131–186.http://dx.doi.Org/10.1353/ajm.2O11.0005 Google Scholar
[FJ14a] Figalli, A. and Jerison, D., Quantitative stability for sumsets in . J. Eur. Math. Soc. 17(2015), no. 5, 1079–1106.http://dx.doi.Org/1 0.41 71/JEMS/527 Google Scholar
[FJ14b] Figalli, A., Quantitative stability for the Brunn-Minkowski inequality. 2014. arxiv:1 502.06513 Google Scholar
[FMP09] Figalli, A., Maggi, F., and Pratelli, A., A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(2009), no. 6, 2511–2519.Google Scholar
[FMP10] Figalli, A., A mass transportation approach to quantitative isoperimetric inequalities. Inv. Math. 182(2010), no. 1, 167–211.http://dx.doi.org/10.1007/s00222-010-0261-z Google Scholar
[FMP08] Fusco, N., Maggi, F., and Pratelli, A., The sharp quantitative isoperimetric inequality. Ann. Math. 168(2008), 941–980.http://dx.doi.Org/10.4007/annals.2008.1 68.941 Google Scholar
[GP69] Gates, D. J. and Penrose, O., The van der Waals limit for classical systems. I. A variational principle. Comm. Math. Phys. 15(1969), 255–276.http://dx.doi.Org/10.1007/BF01645528 Google Scholar
[HLP34] Hardy, G. H., Littlewood, J. E., and Pόlya, G., Inequalities. Cambridge University Press,Cambridge, 1934.Google Scholar
[HM53] Henstock, R. and Macbeath, A. M., On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik. Proc. London Math. Soc. (3) 3(1953), 182–194. http://dx.doi.org/10.1112/plms/s3-3.1.182 Google Scholar
[Lie77] E. H.Lieb, , Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl. Math. 57(1976/77), no. 2, 93–105.http://dx.doi.org/10.1002/sapm197757293 Google Scholar
[LLOl] Lieb, E. H. and Loss, M., Analysis. Second ed., Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.http://dx.doi.Org/10.1090/gsm/014 Google Scholar
[LP66] Lebowitz, J. L. and Penrose, O., Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition. J. Mathematical Phys. 7(1966), 98–113.http://dx.doi.Org/10.1063/1.1704821 Google Scholar
[Magl2] Maggi, F., Sets of finite perimeter and geometric variational problems: an introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics, 135, Cambridge University Press, 2012.http://dx.doi.org/10.1017/CBO9781139108133 Google Scholar
[MN15] Mossel, E. and Neeman, J., Robust dimension free isoperimetry in Gaussian space. Ann. Probab. 43(2015), 971–991. http://dx.doi.org/10.1214/13-AOP860 Google Scholar