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# 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 $\vol(S_{\epsilon})\geq\vol(B_{\epsilon})$ 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.

# References

- 1 Noga Alon and Joel H. Spencer. The probabilistic method. John Wiley & Sons, Inc., second edition, 2000. Zbl 0996.05001.
- 2 Jiří Matoušek. Lectures on Discrete Geometry, volume 212 of GTM. Springer, 2002. Zbl 0999.52006.

## Mathematics Subject Classification

60A10*no label found*51M25

*no label found*51M16

*no label found*

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## Comments

## Encyclopedia entry for "isoperimetric inequality"

I think this entry needs to be re-written somewhat. It contains an undefined term, "figure", in two crucial places (e.g. first sentence). Presumably, continuous "curve" (f : [a, b] --> IR^2, oo \le a < b \le oo) is meant. Also, it is unclear if the inequality is valid for any rectifiable plane curve (including even "curves" continuous only a.e.). In most presentations in books, the curve is assumed to be C^1 or at least piecewise differentiable.

## Re: Encyclopedia entry for "isoperimetric inequality"

i agree about clarifying the term figure. I guess it's no curve, but a compact subset homeomorphic to a disc, so one can take its area and its perimeter (the measure of its boundary).

it would be also very usefull if there mentioned the pages in the books in bibliography where the reader can find more about.