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 -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 (http://planetmath.org/HammingDistance)-like 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. 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.
|Date of creation||2013-03-22 14:02:47|
|Last modified on||2013-03-22 14:02:47|
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