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A singular integral identity for surface measure

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Abstract

We prove that the integral of a certain Riesz-type kernel over \((n-1)\)-rectifiable sets in \({\mathbb {R}}^n\) is constant, from which a formula for surface measure immediately follows. Geometric interpretations are given, and the solution to a geometric variational problem characterizing convex domains follows as a corollary, strengthening a recent inequality of Steinerberger.

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Acknowledgements

My gratitude goes to Stefan Steinerberger for his ideas and suggestions throughout the drafting of this article.

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  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    Ryan E. G. Bushling

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  1. Ryan E. G. Bushling
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Correspondence to Ryan E. G. Bushling.

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Bushling, R.E.G. A singular integral identity for surface measure. J Geom Anal 34, 16 (2024). https://doi.org/10.1007/s12220-023-01443-2

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  • DOI: https://doi.org/10.1007/s12220-023-01443-2

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