isoperimetric inequality

The classical isoperimetric inequality says that if a planar figure has perimeter $P$ and area $A$, then

$$4\pi A\le P^2,$$

where the equality holds if and only if the figure is a circle. That is, the circle is the figure that encloses the largest area among all figures of same perimeter.

The analogous statement is true in arbitrary dimension. The $d$-dimensional ball has the largest volume among all figures of equal surface area.

The isoperimetric inequality can alternatively be stated using the $ϵ$-neighborhoods. An $ϵ$-neighborhood of a set $S$, denoted here by $S_{ϵ}$, is the set of all points whose distance to $S$ is at most $ϵ$. The isoperimetric inequality in terms of $ϵ$-neighborhoods states that $\mathrm{vol}(S_{ϵ})\ge \mathrm{vol}(B_{ϵ})$ where $B$ is the ball of the same volume as $S$. The classical isoperimetric inequality can be recovered by taking the limit $ϵ\to 0$. The advantage of this formulation is that it does not depend on the notion of surface area, and so can be generalized to arbitrary measure spaces with a metric.

An example when this general formulation proves useful is the Talagrand’s isoperimetric theory dealing with Hamming (http://planetmath.org/HammingDistance)-like distances in product spaces. The theory has proven to be very useful in many applications of probability to combinatorics.

Title	isoperimetric inequality
Canonical name	IsoperimetricInequality
Date of creation	2013-03-22 14:02:47
Last modified on	2013-03-22 14:02:47
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	12
Author	bbukh (348)
Entry type	Theorem
Classification	msc 60A10
Classification	msc 51M25
Classification	msc 51M16