# 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
$\u03f5$-neighborhoods^{}. An $\u03f5$-neighborhood of a set $S$,
denoted here by ${S}_{\u03f5}$, is the set of all points whose
distance^{} to $S$ is at most $\u03f5$. The isoperimetric
inequality in terms of $\u03f5$-neighborhoods states that
$\mathrm{vol}({S}_{\u03f5})\ge \mathrm{vol}({B}_{\u03f5})$ where $B$ is the ball of
the same volume as $S$. The classical isoperimetric inequality can
be recovered by taking the limit $\u03f5\to 0$.
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. The probabilistic method. 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 |
---|---|

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 |