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isoperimetric inequality
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(Theorem)
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The classical isoperimetric inequality says that if a planar figure has perimeter $P$ and area $A$ then \begin{equation*} 4\pi A\leq P^2, \end{equation*}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 $\vol(S_\epsilon)\geq \vol(B_\epsilon)$ 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-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.
Zbl 0996.05001.
- 2
- Jirí Matoušek.
Lectures on Discrete Geometry, volume 212 of GTM.
Springer, 2002.
Zbl 0999.52006.
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"isoperimetric inequality" is owned by bbukh.
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Cross-references: applications, product, theory, metric, measure spaces, limit, distance, points, surface area, volume, ball, dimension, circle, equality, area, perimeter, planar
There are 2 references to this entry.
This is version 9 of isoperimetric inequality, born on 2003-10-17, modified 2005-09-17.
Object id is 5397, canonical name is IsoperimetricInequality.
Accessed 5775 times total.
Classification:
| AMS MSC: | 51M16 (Geometry :: Real and complex geometry :: Inequalities and extremum problems) | | | 51M25 (Geometry :: Real and complex geometry :: Length, area and volume) | | | 60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory) |
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Pending Errata and Addenda
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