# isoperimetric inequality

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

 $4\pi A\leq P^{2},$

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 $\epsilon$-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 $\operatorname{vol}(S_{\epsilon})\geq\operatorname{vol}(B_{\epsilon})$ 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 (http://planetmath.org/HammingDistance)-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. . John Wiley & Sons, Inc., second edition, 2000. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0996.05001Zbl 0996.05001.
• 2 Jiří Matoušek. Lectures on Discrete Geometry, volume 212 of GTM. Springer, 2002. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0999.52006Zbl 0999.52006.
Title isoperimetric inequality IsoperimetricInequality 2013-03-22 14:02:47 2013-03-22 14:02:47 bbukh (348) bbukh (348) 12 bbukh (348) Theorem msc 60A10 msc 51M25 msc 51M16