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A New Federer-Type Characterization of Sets of Finite Perimeter

Abstract

Federer’s characterization states that a set \(E\subset \mathbb {R}^n\) is of finite perimeter if and only if \(\mathcal H^{n-1}(\partial ^*E)<\infty \). Here the measure-theoretic boundary \(\partial ^*E\) consists of those points where both E and its complement have positive upper density. We show that the characterization remains true if \(\partial ^*E\) is replaced by a smaller boundary consisting of those points where the lower densities of both E and its complement are at least a given number. This result is new even in Euclidean spaces but we prove it in a more general complete metric space that is equipped with a doubling measure and supports a Poincaré inequality.

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Acknowledgements

The author wishes to thank Nageswari Shanmugalingam for many helpful comments as well as for discussions on constructing spaces where the Mazurkiewicz metric agrees with the ordinary one; Anders Björn also for discussions on constructing such spaces; and Olli Saari for discussions on finding strong boundary points.

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Correspondence to Panu Lahti.

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Lahti, P. A New Federer-Type Characterization of Sets of Finite Perimeter. Arch Rational Mech Anal 236, 801–838 (2020). https://doi.org/10.1007/s00205-019-01483-5

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