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Derivative relationships between volume and surface area of compact regions in R^d

Jean-Luc Marichal·Michael Dorff
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Abstract

We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r = d V/A. We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions, and offer a geometric interpretation of r in a few cases.

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Jean-Luc Marichal, Michael Dorff. 2007-02-22. Derivative relationships between volume and surface area of compact regions in R^d. https://arxiv.org/abs/math/0702635

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