Volume of sublevel sets versus area of level sets via Gelfand-Leray form

TL;DR

This paper establishes a mathematical relation between the volume of sublevel sets and the area of level sets using Gelfand-Leray forms, providing new estimations and proofs for geometric properties.

Contribution

It introduces a novel relation between sublevel set volumes and level set areas via Gelfand-Leray forms, including a proof of a classical geometric fact.

Findings

Derived a relation between sublevel set volume and level set area

Provided an estimation method for sublevel set volume

Proved the derivative of volume of an n-ball equals the surface area of the bounding sphere

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.