isoperimetric inequality

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


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 areaMathworldPlanetmath.

The isoperimetric inequality can alternatively be stated using the ϵ-neighborhoodsMathworldPlanetmath. An ϵ-neighborhood of a set S, denoted here by Sϵ, is the set of all points whose distanceMathworldPlanetmath to S is at most ϵ. The isoperimetric inequality in terms of ϵ-neighborhoods states that vol(Sϵ)vol(Bϵ) where B is the ball of the same volume as S. The classical isoperimetric inequality can be recovered by taking the limit ϵ0. 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 ( distances in product spaces. The theory has proven to be very useful in many applications of probability to combinatorics.


  • 1 Noga Alon and Joel H. Spencer. The probabilistic method. John Wiley & Sons, Inc., second edition, 2000. 0996.05001.
  • 2 Jiří Matoušek. Lectures on Discrete Geometry, volume 212 of GTM. Springer, 2002. 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