The Annals of Probability

Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality

Abstract

We provide a full quantitative version of the Gaussian isoperimetric inequality: the difference between the Gaussian perimeter of a given set and a half-space with the same mass controls the gap between the norms of the corresponding barycenters. In particular, it controls the Gaussian measure of the symmetric difference between the set and the half-space oriented so to have the barycenter in the same direction of the set. Our estimate is independent of the dimension, sharp on the decay rate with respect to the gap and with optimal dependence on the mass.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 668-697.

Dates
Revised: October 2015
First available in Project Euclid: 31 March 2017

https://projecteuclid.org/euclid.aop/1490947306

Digital Object Identifier
doi:10.1214/15-AOP1072

Mathematical Reviews number (MathSciNet)
MR3630285

Zentralblatt MATH identifier
1377.49050

Citation

Barchiesi, Marco; Brancolini, Alessio; Julin, Vesa. Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality. Ann. Probab. 45 (2017), no. 2, 668--697. doi:10.1214/15-AOP1072. https://projecteuclid.org/euclid.aop/1490947306

References

• [1] Acerbi, E., Fusco, N. and Morini, M. (2013). Minimality via second variation for a nonlocal isoperimetric problem. Comm. Math. Phys. 322 515–557.
• [2] Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, New York.
• [3] Bakry, D. and Ledoux, M. (1996). Lévy–Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123 259–281.
• [4] Bobkov, S. G. (1997). An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25 206–214.
• [5] Bögelein, V., Duzaar, F. and Fusco, N. (2015). A quantitative isoperimetric inequality on the sphere. Adv. Calc. Var. To apppear. DOI:10.1515/acv-2015-0042.
• [6] Bögelein, V., Duzaar, F. and Scheven, C. (2015). A sharp quantitative isoperimetric inequality in hyperbolic $n$-space. Calc. Var. Partial Differential Equations 54 3967–4017.
• [7] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
• [8] Brasco, L., De Philippis, G. and Velichkov, B. (2015). Faber–Krahn inequalities in sharp quantitative form. Duke Math. J. 164 1777–1831.
• [9] Carlen, E. A. and Kerce, C. (2001). On the cases of equality in Bobkov’s inequality and Gaussian rearrangement. Calc. Var. Partial Differential Equations 13 1–18.
• [10] Cianchi, A., Fusco, N., Maggi, F. and Pratelli, A. (2011). On the isoperimetric deficit in Gauss space. Amer. J. Math. 133 131–186.
• [11] Cicalese, M. and Leonardi, G. P. (2012). A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206 617–643.
• [12] Ehrhard, A. (1983). Symétrisation dans l’espace de Gauss. Math. Scand. 53 281–301.
• [13] Eldan, R. (2015). A two-sided estimate for the Gaussian noise stability deficit. Invent. Math. 201 561–624.
• [14] Figalli, A., Maggi, F. and Pratelli, A. (2010). A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182 167–211.
• [15] Fusco, N. and Julin, V. (2014). A strong form of the quantitative isoperimetric inequality. Calc. Var. Partial Differential Equations 50 925–937.
• [16] Fusco, N., Maggi, F. and Pratelli, A. (2008). The sharp quantitative isoperimetric inequality. Ann. of Math. (2) 168 941–980.
• [17] Giusti, E. (1984). Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel.
• [18] Maggi, F. (2012). Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics 135. Cambridge Univ. Press, Cambridge.
• [19] McGonagle, M. and Ross, J. (2015). The hyperplane is the only stable, smooth solution to the isoperimetric problem in Gaussian space. Geom. Dedicata 178 277–296.
• [20] Mossel, E. and Neeman, J. (2015). Robust dimension free isoperimetry in Gaussian space. Ann. Probab. 43 971–991.
• [21] Mossel, E. and Neeman, J. (2015). Robust optimality of Gaussian noise stability. J. Eur. Math. Soc. (JEMS) 17 433–482.
• [22] Mossel, E., O’Donnell, R. and Oleszkiewicz, K. (2010). Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. (2) 171 295–341.
• [23] Rosales, C. (2014). Isoperimetric and stable sets for log-concave perturbations of Gaussian measures. Anal. Geom. Metr. Spaces 2 328–358.
• [24] Sternberg, P. and Zumbrun, K. (1998). A Poincaré inequality with applications to volume-constrained area-minimizing surfaces. J. Reine Angew. Math. 503 63–85.
• [25] Sudakov, V. N. and Tsirelson, B. S. (1974). Extremal properties of half-spaces for spherically invariant measures: Problems in the theory of probability distributions, II. Zap. NauČn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 14–24, 165.
• [26] Tamanini, I. (1984). Regularity results for almost minimal oriented hypersurfaces in $\mathbb{R}^{n}$. Quaderni del Dipartimento di Matematica Dell’Università di Lecce, Lecce. Available at cvgmt.sns.it/paper/1807/.
• [27] Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics 140. Amer. Math. Soc., Providence, RI.
• [28] Villani, C. (2009). Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.