Editing Isoperimetric inequality (section)

== In Euclidean space  ==
The isoperimetric inequality states that a [[sphere]] has the smallest surface area per given volume. Given a bounded open set <math>S\subset\R ^n</math> with <math>C^1</math> boundary, having [[surface area]] <math>\operatorname{per}(S)</math> and [[volume]] <math>\operatorname{vol}(S)</math>, the isoperimetric inequality states

:<math>\operatorname{per}(S)\geq n \operatorname{vol}(S)^{(n-1)/n} \, \operatorname{vol}(B_1)^{1/n},</math>

where <math>B_1\subset\R ^n</math> is a [[Unit sphere|unit ball]]. The equality holds when <math>S</math> is a ball in <math>\R ^n</math>. Under additional restrictions on the set (such as [[Convex set|convexity]], [[Closed regular set|regularity]], [[Smooth surface|smooth boundary]]), the equality holds for a ball only. But in full generality the situation is more complicated. The relevant result of {{harvtxt|Schmidt|1949|loc=Sect. 20.7}} (for a simpler proof see {{harvtxt|Baebler|1957}}) is clarified in {{harvtxt|Hadwiger|1957|loc=Sect. 5.2.5}} as follows. An extremal set consists of a ball and a "corona" that contributes neither to the volume nor to the surface area. That is, the equality holds for a compact set <math>S</math> if and only if <math>S</math> contains a closed ball <math>B</math> such that <math>\operatorname{vol}(B) = \operatorname{vol}(S)</math> and <math>\operatorname{per}(B) = \operatorname{per}(S).</math> For example, the "corona" may be a curve.


The proof of the inequality follows directly from [[Brunn–Minkowski theorem|Brunn–Minkowski inequality]] between a set <math>S</math> and a ball with radius <math>\epsilon</math>, i.e. <math>B_\epsilon=\epsilon B_1</math>. Indeed, <math>\operatorname{vol}(A + B_\epsilon) \ge (\operatorname{vol}(A)^{1/n} + \operatorname{vol}(B_\epsilon)^{1/n})^n \ge \operatorname{vol}(A) + n \operatorname{vol}(A)^{(n-1)/n} \epsilon\operatorname{vol}(B_1)^{1/n}.</math> The isoperimetric inequality follows by subtracting <math  display="inline">\operatorname{vol}(A)</math>, dividing by <math>\epsilon</math>, and taking the limit as <math>\epsilon\to 0.</math> ({{harvtxt|Osserman|1978}}; {{harvtxt|Federer|1969|loc=§3.2.43}}).

In full generality {{harv|Federer|1969|loc=§3.2.43}}, the isoperimetric inequality states that for any set <math>S\subset\R^n</math> whose [[closure of a set|closure]] has finite [[Lebesgue measure]]

:<math>n\,\omega_n^{1/n} L^n(\bar{S})^{(n-1)/n} \le M^{n-1}_*(\partial S)</math>

where <math>M_*^{n-1}</math> is the (''n''-1)-dimensional [[Minkowski content]], ''L<sup>n</sup>'' is the ''n''-dimensional Lebesgue measure, and ''ω<sub>n</sub>'' is the volume of the [[unit ball]] in <math>\R^n</math>. If the boundary of ''S'' is [[rectifiable set|rectifiable]], then the Minkowski content is the (''n''-1)-dimensional [[Hausdorff measure]].

The ''n''-dimensional isoperimetric inequality is equivalent (for sufficiently smooth domains) to the [[Sobolev inequality]] on <math>\R^n</math> with optimal constant:

:<math>\left( \int_{\R^n} |u|^{n/(n-1)}\right)^{(n-1)/n} \le n^{-1}\omega_{n}^{-1/n}\int_{\R^n}|\nabla u|</math>

for all <math>u\in W^{1,1}(\R^n)</math>.