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Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory by Francesco Maggi

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18

Slicing boundaries of sets of finite perimeter

In this chapter we consider the problem of computing the perimeter of a set as an integral of the Hausdorff measures of the lower dimensional sections (through a given slicing function) of its boundary. The basic tool here is the extension of the coarea formula to rectifiable sets (Section 18.2), which in turn is based on a revised version of the coarea formula (13.1) (Section 18.1). In Section 18.3 we study slicing by hyperplanes, which will prove to be a useful technical tool in the study of sessile liquid drops (Chapter 19), as well as in the regularity theory of Part III; see, in particular, the proof of Theorem 22.8. In fact, in this last context, we shall also apply slicing by cylinders ...

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