Abstract

We establish a sharp form of a nonlocal quantitative isoperimetric inequality involving the barycentric asymmetry for convex sets. This result can be seen as the nonlocal analogue of the one obtained by B. Fuglede [Lower estimate of the isoperimetric deficit of convex domains in Rn\mathbf{R}^n in terms of asymmetry, Geom. Dedicata 47/1 (1993) 41–48]. A main tool in the proof is an estimate from below of the fractional perimeter by a negative power of the inradius for convex sets

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Fractional perimeter
  • quantitative isoperimetric inequality
  • barycentric asymmetry

Mathematics Subject Classification

52A38, 49K40, 52A05; 28A75, 49Q10

References

  1. F. Bernstein: Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene, Math. Ann. 60/1(1905) 117–136.https://doi.org/10.1007/BF01447496Google Scholar
  2. C. Bianchini, G. Croce, A. Henrot: On the quantitative isoperimetric inequality in the plane with the barycentric asymmetry, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 24/4 (2023) 2477–2500.https://doi.org/10.2422/2036-2145.202107_014Google Scholar
  3. T. Bonnesen: Über das isoperimetrische Defizit ebener Figuren, Math. Ann. 91/3-4 (1924) 252–268.https://doi.org/10.1007/BF01556082Google Scholar
  4. J. Bourgain, H. Brezis, P. Mironescu: Another look at Sobolev spaces, in: Optimal Control and Partial Differential Equations, J. L. Menaldi et al. (eds.), Proc. Conf. Paris 2000, IOS Press, Amsterdam (2001) 439–455.Google Scholar
  5. C. Bucur, E. Valdinoci: Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana Vol. 20, Springer, Cham (2016).https://doi.org/10.1007/978-3-319-28739-3Google Scholar
  6. L. Caffarelli, J.-M. Roquejoffre, O. Savin: Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63/9 (2010) 1111–1144.https://doi.org/10.1002/cpa.20331Google Scholar
  7. D. Carazzato, G. Pascale, M. Pozzetta: Quantitative isoperimetric inequalities in capillarity problems and cones in strong and barycentric forms, arXiv: 2507.07686 (2025).Google Scholar
  8. M. Cicalese, G. P. Leonardi: A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Analysis 206/2 (2012) 617–643.https://doi.org/10.1007/s00205-012-0544-1Google Scholar
  9. J. Dávila: On an open question about functions of bounded variation, Calc. Var. Partial Diff. Equations 15 (2002) 519–527.https://doi.org/10.1007/s005260100135Google Scholar
  10. A. Di Castro, M. Novaga, B. Ruffini, E. Valdinoci: Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Diff. Equations 54/3 (2015) 2421–2464.https://doi.org/10.1007/s00526-015-0870-xGoogle Scholar
  11. E. Di Nezza, G. Palatucci, E. Valdinoci: Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136/5 (2012) 521–573.https://doi.org/10.1016/j.bulsci.2011.12.004Google Scholar
  12. A. Figalli, N. Fusco, F. Maggi, V. Millot, M. Morini: Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys. 336/1 (2015) 441–507.https://doi.org/10.1007/s00220-014-2244-1Google Scholar
  13. A. Figalli, F. Maggi, A. Pratelli: A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182/1 (2010) 167–211.https://doi.org/10.1007/s00222-010-0261-zGoogle Scholar
  14. R. L. Frank, R. Seiringer: Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Analysis 255/12 (2008) 3407–3430.https://doi.org/10.1016/j.jfa.2008.05.015Google Scholar
  15. B. Fuglede: Stability in the isoperimetric problem for convex or nearly spherical domains in Rn, Trans. Amer. Math. Soc. 314/2 (1989) 619–638.https://doi.org/10.2307/2001401Google Scholar
  16. B. Fuglede: Lower estimate of the isoperimetric deficit of convex domains in Rn in terms of asymmetry, Geom. Dedicata 47/1 (1993) 41–48.https://doi.org/10.1007/BF01263492Google Scholar
  17. N. Fusco: The quantitative isoperimetric inequality and related topics, Bull. Math. Sci. 5/3 (2015) 517–607.https://doi.org/10.1007/s13373-015-0074-xGoogle Scholar
  18. N. Fusco, F. Maggi, A. Pratelli: The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168/3 (2008) 941–980.https://doi.org/10.4007/annals.2008.168.941Google Scholar
  19. C. Gambicchia, A. Pratelli: The sharp quantitative barycentric isoperimetric inequality for bounded sets, Ann. Scuola Norm. Sup., Cl. Scienze 15 (2025).https://doi.org/10.2422/2036-2145.202407_013Google Scholar
  20. R. R. Hall: A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428(1992) 161–176.https://doi.org/10.1515/crll.1992.428.161Google Scholar
  21. L. Lombardini: Fractional perimeters from a fractal perspective, Adv. Nonlinear Studies 19/1 (2019) 165–196.https://doi.org/10.1515/ans-2018-2016Google Scholar
  22. V. Maz’ya, T. Shaposhnikova: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Analysis 195/2 (2002) 230–238.https://doi.org/10.1006/jfan.2002.3955Google Scholar
  23. A. C. Ponce, D. Spector: A note on the fractional perimeter and interpolation, C. R. Math. Acad. Sci. Paris 355/9 (2017) 960–965.https://doi.org/10.1016/j.crma.2017.09.001Google Scholar
  24. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series Vol. 28, Princeton University Press, Princeton (1970).Google Scholar
  25. R. Schneider: Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications Vol. 151, expanded edition, Cambridge University Press, Cambridge (2014).Google Scholar

Suggested citation

C. Gambicchia, E. M. Merlino, B. Ruffini, M. Talluri. “Barycentric Stability of Nonlocal Perimeters: the Convex Case.” Journal of Convex Analysis 34 (2027), No. 1, 144–153. https://doi.org/10.68381/jca34011

Download BibTeXPublished by Banach Press.

© 2026 Banach Press LLC. Published under CC BY 4.0.