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Publisher: Elsevier
Languages: English
Types: Article
Subjects: QA299, 28A75, 49Q15, Mathematics - Metric Geometry
We study integralgeometric representations of variations of general sets $A$ in the Euclidean n-space without any regularity assumptions. If we assume, for example, that just one partial derivative of its characteristic function $\chi^A$ is a signed Borel measure with finite total variation, can we provide a nice integralgeometric representation of this variation? This is a delicate question, as the Gauss-Green type theorems of De Giorgi and Federer are not available in this generality. We will show that a `measure-theoretic boundary' plays its role in such representations similarly as for the sets of finite variation. There is a variety of suitable notions of `measure-theoretic boundary' and one can address the question to find notions of measure-theoretic boundary that are as fine as possible.
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