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

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

The isoperimetric inequality can alternatively be stated using the -neighborhoods. An -neighborhood of a set , denoted here by , is the set of all points whose distance to is at most . The isoperimetric inequality in terms of -neighborhoods states that where is the ball of the same volume as . The classical isoperimetric inequality can be recovered by taking the limit . 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

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