Perimeter

2020 Mathematics Subject Classification: Primary: 26B15 [MSN][ZBL]

The perimeter of a planar region bounded by a rectifiable curve is the total length of the corresponding curve. In higher dimension, the perimeter of an open set $U\subset \mathbb R^n$ with $C^1$ boundary $\partial U$ (i.e. such that $\partial U$ is a $C^1$ submanifold) is the $n-1$-dimensional volume of $\partial U$, namely \[ {\rm Vol}^{n-1} (\partial U) = \int_{\partial U} {\rm d vol} \] where ${\rm d vol}$ is the $n-1$ dimensional volume form, for the Riemannian structure induced by the restriction of the scalar Euclidean product on the tangent space to $\partial U$. In fact such integral coincides with the $n-1$-dimensional Hausdorff measure of $\partial U$, see Area formula. An analogous definition can be given for open subsets of a Riemannian manifold with $C^1$ boundary.

For Lebesgue measurable subsets of $\mathbb R^n$ a very general definition was proposed originally by Caccioppoli in [Ca] and used later to build a far-reaching theory by De Giorgi (see for instance [Gi]).

Definition 1 Let $E\subset \mathbb R^n$. The perimeter of $E$ is defined as the infimum of \[ \liminf_{k\to \infty} {\rm Vol}^{n-1} (\partial U_k) \] taken over all sequences of open sets $U_k$ with smooth boundary such that $\lambda ((U_k\setminus E) \cup (E \setminus U_k)) \to 0$ (here $\lambda$ denotes the $n$-dimensional Lebesgue measure).

This notion of perimeter coincides with the usual one if, for instance, $E$ is an open set with piecewise smooth boundary.

Remark 2 Actually, in the original definition of Caccioppoli and De Giorgi the infimum is taken over sequences of polytopes, where ${\rm Vol}^{n-1} (\partial U_k)$ is defined as the sum of the $n-1$-dimensional volumes of the corresponding faces. The definition given above is however much more convenient, it is the most common in modern textbooks and it is equivalent to the original one of Caccioppoli.

A set $E$ for which the perimeter is finite is called set of finite perimeter or Caccioppoli set. A fundamental characterization, due to De Giorgi is the following

Theorem 3 A measurable set $E$ with $\lambda (E) < \infty$ has finite perimeter if and only if the indicator function \[ {\bf 1}_E (x):= \left\{\begin{array}{ll} 1\qquad & \mbox{if } x\in E\\ 0 & \mbox{otherwise} \end{array} \right. \] is a function of bounded variation.

See Function of bounded variation for the most important properties of the sets of finite perimeter.

References

[Be]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[BZ]	Yu.D. Burago, V.A. Zalgaller, "Geometric inequalities" , Springer (1988) (Translated from Russian)
[Ca]	R. Caccioppoli, "Misura e integrazione sugli insiemi dimensionalmente orientati I" Rend. Accad. Naz. Lincei Ser. 8 , 12 : 1 (1952) pp. 3–11
[Ca1]	R. Caccioppoli, "Misura e integrazione sugli insiemi dimensionalmente orientati II" Rend. Accad. Naz. Lincei Ser. 8 , 12 : 2 (1952) pp. 137–146
[DG]	E. de Giorgi, "Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita" Rend. Accad. Naz. Lincei Ser. 1 , 5 : 2 (1958) pp. 33–34
[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]	H. Federer, "Geometric measure theory", Springer-Verlag (1979). MR0257325 Zbl 0874.49001
[Gi]	E. Giusti, "Minimal surfaces and functions of bounded variation" , Birkhäuser (1984)
[Si]	L. Simon, "Lectures on geometric measure theory", Proceedings of the Centre for Mathematical Analysis, 3. Australian National University. Canberra (1983) MR0756417 Zbl 0546.49019
[Sp]	M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965) MR0209411 Zbl 0141.05403