# Perimeter

2010 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.

How to Cite This Entry:
Perimeter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perimeter&oldid=30848
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article