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


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.

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


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

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Correspondence to Duc Tai Trinh.

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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

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

Mathematics Subject Classification (2010)

  • 58A10
  • 26B15