isoperimetric inequality

The classical isoperimetric inequality says that if a planarfigure 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 areaamong all figures of same perimeter.

The analogous statement is true in arbitrary dimension. The$d$-dimensional ball has the largest volume among all figures ofequal 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 whosedistance to $S$ is at most $ϵ$. The isoperimetricinequality in terms of $ϵ$-neighborhoods states that$\mathrm{vol}(S_{ϵ})\ge \mathrm{vol}(B_{ϵ})$ where $B$ is the ball ofthe same volume as $S$. The classical isoperimetric inequality canbe recovered by taking the limit $ϵ\to 0$.The advantage of this formulation is that itdoes not depend on the notion of surface area, and so can begeneralized to arbitrary measure spaces with a metric.

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