Abstract
We prove that balls centered at the origin and with small radius are stable local minimizers of the Gaussian perimeter among all symmetric sets. Precisely, using the second variation of the Gaussian perimeter, we show that if the radius is smaller than
References
[1] E. Acerbi, N. Fusco and M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Comm. Math. Phys. 322 (2013), no. 2, 515–557. 10.1007/s00220-013-1733-ySuche in Google Scholar
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Math. Monogr., Clarendon Press, New York, 2000. 10.1093/oso/9780198502456.001.0001Suche in Google Scholar
[3] M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, Ann. Probab. 45 (2017), no. 2, 668–697. 10.1214/15-AOP1072Suche in Google Scholar
[4] F. Barthe, An isoperimetric result for the Gaussian measure and unconditional sets, Bull. Lond. Math. Soc. 33 (2001), no. 4, 408–416. 10.1017/S0024609301008141Suche in Google Scholar
[5] V. Bögelein, F. Duzaar and N. Fusco, A sharp quantitative isoperimetric inequality in higher codimension, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 3, 309–362. 10.4171/RLM/709Suche in Google Scholar
[6] V. Bögelein, F. Duzaar and N. Fusco, A quantitative isoperimetric inequality on the sphere, Adv. Calc. Var. 10 (2017), no. 3, 223–265. 10.1515/acv-2015-0042Suche in Google Scholar
[7] C. Borell, The Brunn–Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207–216. 10.1007/BF01425510Suche in Google Scholar
[8] L. Brasco, G. De Philippis and B. Velichkov, Faber–Krahn inequalities in sharp quantitative form, Duke Math. J. 164 (2015), no. 9, 1777–1831. 10.1215/00127094-3120167Suche in Google Scholar
[9] E. A. Carlen and C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations 13 (2001), no. 1, 1–18. 10.1007/PL00009921Suche in Google Scholar
[10] A. Chakrabarti and O. Regev, An optimal lower bound on the communication complexity of gap-Hamming-distance, STOC’11 – Proceedings of the 43rd ACM Symposium on Theory of Computing, ACM, New York (2011), 51–60. 10.1145/1993636.1993644Suche in Google Scholar
[11] A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, On the isoperimetric deficit in Gauss space, Amer. J. Math. 133 (2011), no. 1, 131–186. 10.1353/ajm.2011.0005Suche in Google Scholar
[12] M. Cicalese and G. P. Leonardi, A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Anal. 206 (2012), no. 2, 617–643. 10.1007/s00205-012-0544-1Suche in Google Scholar
[13] R. Eldan, A two-sided estimate for the Gaussian noise stability deficit, Invent. Math. 201 (2015), no. 2, 561–624. 10.1007/s00222-014-0556-6Suche in Google Scholar
[14] N. Fusco, The quantitative isoperimetric inequality and related topics, Bull. Math. Sci. 5 (2015), no. 3, 517–607. 10.1007/s13373-015-0074-xSuche in Google Scholar
[15] N. Fusco and V. Julin, A strong form of the quantitative isoperimetric inequality, Calc. Var. Partial Differential Equations 50 (2014), no. 3–4, 925–937. 10.1007/s00526-013-0661-1Suche in Google Scholar
[16] N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008), no. 3, 941–980. 10.4007/annals.2008.168.941Suche in Google Scholar
[17] S. Heilman, Low correlation noise stability of symmetric sets, preprint (2016), https://arxiv.org/abs/1511.00382v4. 10.1007/s10959-020-01031-ySuche in Google Scholar
[18] S. Heilman, Symmetric convex sets with minimal Gaussian surface area, preprint (2017), https://arxiv.org/abs/1705.06643v1. 10.1353/ajm.2021.0000Suche in Google Scholar
[19] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Stud. Adv. Math. 135, Cambridge University Press, Cambridge, 2012. 10.1017/CBO9781139108133Suche in Google Scholar
[20] E. Mossel and J. Neeman, Robust dimension free isoperimetry in Gaussian space, Ann. Probab. 43 (2015), no. 3, 971–991. 10.1214/13-AOP860Suche in Google Scholar
[21] E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: Invariance and optimality, Ann. of Math. (2) 171 (2010), no. 1, 295–341. 10.4007/annals.2010.171.295Suche in Google Scholar
[22] R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014. 10.1017/CBO9781139814782Suche in Google Scholar
[23] V. N. Sudakov and B. S. Cirel’son, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14–24, 165. 10.1007/BF01086099Suche in Google Scholar
[24] I. Tamanini, Regularity Results for Almost Minimal Oriented Hypersurfaces, Quaderni del Dipartimento di Matematica dell’Università di Lecce, Lecce, 1984; see also http://cvgmt.sns.it/paper/1807/. Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Morrey spaces and generalized Cheeger sets
- Heat ball formulæ for k-forms on evolving manifolds
- A phase-field approximation of the Steiner problem in dimension two
- Maximal solutions for the ∞-eigenvalue problem
- Local minimality of the ball for the Gaussian perimeter
- A direct approach to the anisotropic Plateau problem
Artikel in diesem Heft
- Frontmatter
- Morrey spaces and generalized Cheeger sets
- Heat ball formulæ for k-forms on evolving manifolds
- A phase-field approximation of the Steiner problem in dimension two
- Maximal solutions for the ∞-eigenvalue problem
- Local minimality of the ball for the Gaussian perimeter
- A direct approach to the anisotropic Plateau problem