Volume of Sublevel Sets Versus Area of Level Sets via Gelfand-Leray Form

Abstract

In this paper, we give a relation between the volume of sublevel sets and the area of level sets using a Gelfand-Leray form. As a consequence, we give an estimation of the volume of sublevel sets. In particular, we give a proof of the known fact that the derivative of the volume of a n-dimensional ball with respect to the radius equals the area of the sphere which bounds the n-dimensional ball.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. 1.

    Arnol’d, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. Vol. II, vol. 83. Birkhäuser Boston, Inc, Boston (1988)

    Book  Google Scholar 

  2. 2.

    Carbery, A., Christ, M., Wright, J.: Multidimensional van der Corput and sublevel set estimates. J. Am. Math. Soc. 12(4), 981–1015 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Carbery, A., Wright, J.: What is van der Corput’s lemma in higher dimensions. Publ. Mat. Vol. Extra 13–26 (2002)

  4. 4.

    Leray, J.: Le calcul différentiel et intégral sur une variété analytique complexe. Bull. Soc. Math. France 87, 81–180 (1959)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Łojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris (1962), Editions du C.N.R.S., Paris, pp. 87–89 (1963)

  6. 6.

    Phong, D.H., Stein, E.M., Sturm, J.: Multilinear level set operators, oscillatory integral operators, and Newton polyhedra. Math. Ann. 319(3), 573–596 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43. Princeton University Press, Princeton (2016)

    Google Scholar 

  8. 8.

    Tu, L.W.: An Introduction to Manifolds. (Universitext). Springer, New York (2008)

    Google Scholar 

  9. 9.

    Vasil’ev, B.A.: The asymptotic behavior of exponential integrals, the Newton’s diagram and the classification of minima. Funktcional. Anal. i Priložen 11(3), 1–11 (1977)

    MathSciNet  Google Scholar 

  10. 10.

    Yomdin, Y., Comte, G.: Tame geometry with application in smooth analysis. In: Lecture Notes in Mathematics, 1834. Springer, Berlin (2004)

Download references

Acknowledgements

We would like to thank the referees for their helpful suggestions and comments on the paper.

Funding

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.324.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Duc Tai Trinh.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Trinh, D.T. Volume of Sublevel Sets Versus Area of Level Sets via Gelfand-Leray Form. Acta Math Vietnam 44, 915–922 (2019). https://doi.org/10.1007/s40306-019-00332-7

Download citation

Keywords

  • Differential form
  • Gelfand-Leray form
  • Sublevel sets
  • Volume and area

Mathematics Subject Classification (2010)

  • 58A10
  • 26B15