6 feb 2017

Levico Terme

**Abstract.**

We consider the new characterization of the perimeter of a measurable set in $\mathbb{R}^{n}$ recently studied by Ambrosio, Bourgain, Brezis and Figalli. We modify their approach by using, instead of cubes, covering families made by translations of a given open bounded set with Lipschitz boundary. By adapting the argument of the proof of the differentiation theorem for measures to nondecreasing superadditive or subadditive set functionals, we show that the functionals defined through the new coverings converge to an anisotropic surface measure. Such a measure is a multiple of the perimeter if we allow for isotropic coverings (e.g. balls or arbitrary rotations of the given set).

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