Abstract
We introduce a new variational method for the study of isoperimetric inequalities with quantitative terms. The method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two notable applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in \({\mathbb{R}^n}\). Second we positively answer a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in \({\mathbb{R}^2}\) in the small asymmetry regime.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Almgren F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4, viii+199 (1976)
Alvino A., Ferone V., Nitsch C.: A sharp isoperimetric inequality in the plane. J. Eur. Math. Soc. 13, 185–206 (2011)
Ambrosio, L.: Corso introduttivo alla teoria geometrica della misura ed alle superfici minime. Appunti dei Corsi Tenuti da Docenti della Scuola. [Notes of Courses Given by Teachers at the School]. Scuola Normale Superiore, Pisa, 1997
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000
Bernstein F.: Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60, 117–136 (1905)
Bonnesen T.: Über eine Verschärfung der isoperimetrischen Ungleichheit des Kreises in der Ebene und auf der Kugeloberfläche nebst einer Anwendung auf eine Minkowskische Ungleichheit für konvexe Körper. Math. Ann. 84, 216–227 (1921)
Bonnesen T.: Über das isoperimetrische Defizit ebener Figuren. Math. Ann. 91, 252–268 (1924)
Cianchi A., Fusco N., Maggi F., Pratelli A.: On the isoperimetric deficit in Gauss space. Am. J. Math. 133, 131–186 (2011)
Cicalese, M., Leonardi, G.: Best constants for the quantitative isoperimetric inequality. J. Eur. Math. Soc. (to appear)
De Giorgi, E.: Frontiere orientate di misura minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–1961. Editrice Tecnico Scientifica, Pisa, 1961
Dinghas A.: Bemerkung zu einer Verschärfung der isoperimetrischen Ungleichung durch H. Hadwiger. Math. Nachr. 1, 284–286 (1948)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1992
Figalli A., Maggi F., Pratelli A.: A note on Cheeger sets. Proc. Am. Math. Soc. 137, 2057–2062 (2009)
Figalli A., Maggi F., Pratelli A.: A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2511–2519 (2009)
Figalli A., Maggi F., Pratelli A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167–211 (2010)
Fuglede B.: Stability in the isoperimetric problem. Bull. Lond. Math. Soc. 18, 599–605 (1986)
Fuglede B.: Stability in the isoperimetric problem for convex or nearly spherical domains in R n. Trans. Am. Math. Soc. 314, 619–638 (1989)
Fusco N., Maggi F., Pratelli A.: The sharp quantitative Sobolev inequality for functions of bounded variation. J. Funct. Anal. 244, 315–341 (2007)
Fusco N., Maggi F., Pratelli A.: The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168, 941–980 (2008)
Fusco N., Maggi F., Pratelli A.: Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8, 51–71 (2009)
Giusti, E.: Minimal surfaces and functions of bounded variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel, 1984
Hadwiger H.: Die isoperimetrische Ungleichung im Raum. Elemente der Math. 3, 25–38 (1948)
Hall R.R.: A quantitative isoperimetric inequality in n-dimensional space. J. Reine Angew. Math. 428, 161–176 (1992)
Hall R.R., Hayman W.K.: A problem in the theory of subordination. J. Anal. Math. 60, 99–111 (1993)
Hall R.R., Hayman W.K., Weitsman A.W.: On asymmetry and capacity. J. Anal. Math. 56, 87–123 (1991)
Maggi F.: Some methods for studying stability in isoperimetric type problems. Bull. Am. Math. Soc. (N.S.) 45, 367–408 (2008)
Massari U.: Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in R n. Arch. Rational Mech. Anal. 55, 357–382 (1974)
Milman, V.D., Schechtman, G.: Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, vol. 1200. Springer, Berlin, 1986
Morgan F.: Clusters minimizing area plus length of singular curves. Math. Ann. 299, 697–714 (1994)
Morgan F., Ros A.: Stable constant-mean-curvature hypersurfaces are area minimizing in small L 1 neighborhoods. Interfaces Free Bound. 12, 151–155 (2010)
Osserman R.: Bonnesen-style isoperimetric inequalities. Amer. Math. Monthly 86, 1–29 (1979)
Tamanini I.: Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334, 27–39 (1982)
Tamanini I.: Regularity results for almost minimal oriented hypersurfaces in R n. Quaderni del Dipartimento di Matematica dell’ Università di Lecce 1, 1–92 (1984)
White B.: A strong minimax property of nondegenerate minimal submanifolds. J. Reine Angew. Math. 457, 203–218 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Müller
Rights and permissions
About this article
Cite this article
Cicalese, M., Leonardi, G.P. A Selection Principle for the Sharp Quantitative Isoperimetric Inequality. Arch Rational Mech Anal 206, 617–643 (2012). https://doi.org/10.1007/s00205-012-0544-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-012-0544-1