The classical isoperimetric inequality says that if a planar figure has perimeter $P$ and area $A$ then \begin{equation*} 4\pi A\leq P^2, \end{equation*}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 $\epsilon$ neighborhoods. An $\epsilon$ neighborhood of a set $S$ denoted here by $S_\epsilon$ is the set of all points whose distance to $S$ is at most $\epsilon$ The isoperimetric inequality in terms of $\epsilon$ neighborhoods states that $\vol(S_\epsilon)\geq \vol(B_\epsilon)$ where $B$ is the ball of the same volume as $S$ The classical isoperimetric inequality can be recovered by taking the limit $\epsilon\to 0$ 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-like distances in product spaces. The theory has proven to be very useful in many applications of probability to combinatorics.

References

1

Noga Alon and Joel H. Spencer.
The probabilistic method.
John Wiley & Sons, Inc., second edition, 2000.
Zbl 0996.05001.

2

Jirí Matoušek.
Lectures on Discrete Geometry, volume 212 of GTM.
Springer, 2002.
Zbl 0999.52006.