Perimeter of a set

Summary

The perimeter of a measurable set $E \subseteq \R^n$ is a measure-theoretic version of the area of its boundary. It is the total variation of the distributional gradient of the indicator function of $E$. For a set with smooth boundary it equals the $(n-1)$-dimensional area of the boundary, but it is defined for every Lebesgue measurable set. It is lower semicontinuous under convergence in $L^1_{\mathrm{loc}}$, which makes it the natural quantity to minimize in problems such as the isoperimetric problem.