Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

IMPACT FACTOR 2018: 2.316

CiteScore 2018: 1.77

SCImago Journal Rank (SJR) 2018: 2.350
Source Normalized Impact per Paper (SNIP) 2018: 1.465

Mathematical Citation Quotient (MCQ) 2017: 1.15

See all formats and pricing
More options …
Volume 10, Issue 3


A quantitative isoperimetric inequality on the sphere

Verena Bögelein / Frank Duzaar / Nicola Fusco
  • Corresponding author
  • Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Napoli, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-01-28 | DOI: https://doi.org/10.1515/acv-2015-0042


In this paper we prove a quantitative version of the isoperimetric inequality on the sphere with a constant independent of the volume of the set E.

Keywords: Isoperimetric inequality; stability; sphere

MSC 2010: 49Q20; 58E35


  • [1]

    E. Acerbi, N. Fusco and M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Comm. Math. Phys. 322 (2013), 515–557. Google Scholar

  • [2]

    F. Almgren, Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35 (1986), no. 3, 451–547. Google Scholar

  • [3]

    M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, preprint (2015), http://http://arxiv.org/abs/1409.2106v2.

  • [4]

    M. Barchiesi, F. Cagnetti and N. Fusco, Stability of the Steiner symmetrization of convex sets, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1245–1278. Google Scholar

  • [5]

    F. Bernstein, Über die isoperimetriche Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene, Math. Ann. 60 (1905), 117–136. Google Scholar

  • [6]

    V. Bögelein, F. Duzaar and N. Fusco, A sharp quantitative isoperimetric inequality in higher codimension, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2012), no. 3, 309–362. Google Scholar

  • [7]

    T. Bonnesen, Über das isoperimetrische Defizit ebener Figuren, Math. Ann. 91 (1924), 252–268. Google Scholar

  • [8]

    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. Web of ScienceGoogle Scholar

  • [9]

    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. Web of ScienceGoogle Scholar

  • [10]

    M. Cicalese, G. P. Leonardi and F. Maggi, Sharp stability inequalities for planar double bubbles, preprint (2015), http://arxiv.org/abs/1211.3698v2.

  • [11]

    G. David and S. Semmes, Quasiminimal surfaces of codimension 1 and John domains, Pacific J. Math. 183 (1998), no. 2, 213–277. Google Scholar

  • [12]

    F. Duzaar and K. Steffen, Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds, Indiana Univ. Math. J. 45 (1996), no. 4, 1045–1093. Google Scholar

  • [13]

    F. Duzaar and K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002), 73–138. Google Scholar

  • [14]

    A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182 (2010), 167–211. Google Scholar

  • [15]

    B. Fuglede, Stability in the isoperimetric problem for convex of nearly spherical domains in n, Trans. Amer. Math. Soc. 314 (1989), no. 2, 619–638. Google Scholar

  • [16]

    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. Google Scholar

  • [17]

    N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008), no. 3, 941–980. Google Scholar

  • [18]

    R. R. Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428 (1992), 161–176. Google Scholar

  • [19]

    R. R. Hall, W. K. Hayman and A. W. Weitsman, On asymmetry and capacity, J. Anal. Math. 56 (1991), 87–123. Google Scholar

  • [20]

    J. Kinnunen, R. Korte, A. Lorent and N. Shanmugalingam, Regularity of sets with quasiminimal boundary surfaces in metric spaces, J. Geom. Anal. 23 (2013), 1607–1640. Web of ScienceGoogle Scholar

  • [21]

    F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Stud. Adv. Math. 135, Cambridge University Press, Cambridge, 2012. Google Scholar

  • [22]

    E. Schmidt, Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionszahl, Math. Z. 49 (1943/44), 1–109. Google Scholar

  • [23]

    L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. Google Scholar

  • [24]

    I. Tamanini, Regularity results for almost minimal oriented hypersurfaces in n, Quad. Dipart. Mat. Univ. Lecce 1 (1984), 1–92. Google Scholar

  • [25]

    B. White, A strong minimax property of nondegenerate minimal submanifolds, J. Reine Angew. Math. 457 (1994), 203–218. Google Scholar

About the article

Received: 2015-11-05

Accepted: 2015-12-01

Published Online: 2016-01-28

Published in Print: 2017-07-01

This research was supported by the 2008 ERC Advanced Grant “Analytic Techniques for Geometric and Functional Inequalities” (Grant no. 226234 AnTeGeFI). The research of the third author was partially carried on at the University of Jyväskylä in the framework of the “FiDiPro” program of the Finnish Academy of Science (no. 2100002028).

Citation Information: Advances in Calculus of Variations, Volume 10, Issue 3, Pages 223–265, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2015-0042.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in