Editing Pi (section)

=== Inequalities ===
[[File:Sir William Thompson illustration of Carthage.png|thumb|The [[ancient Carthage|ancient city of Carthage]] was the solution to an isoperimetric problem, according to a legend recounted by [[Lord Kelvin]]:<ref>{{cite journal |first=William |last=Thompson |author-link=Lord Kelvin |title=Isoperimetrical problems |year=1894 |journal=Nature Series: Popular Lectures and Addresses |volume=II |pages=571–592}}</ref> those lands bordering the sea that [[Dido|Queen Dido]] could enclose on all other sides within a single given oxhide, cut into strips.|left]]

The number {{pi}} serves appears in similar eigenvalue problems in higher-dimensional analysis.  As mentioned [[#Definition|above]], it can be characterized via its role as the best constant in the [[isoperimetric inequality]]: the area {{mvar|A}} enclosed by a plane [[Jordan curve]] of perimeter {{mvar|P}} satisfies the inequality

<math display=block>4\pi A\le P^2,</math>

and equality is clearly achieved for the circle, since in that case {{math|1=''A'' = π''r''{{sup|2}}}} and {{math|1=''P'' = 2π''r''}}.<ref>{{cite book |first=Isaac |last=Chavel |title=Isoperimetric inequalities |publisher=Cambridge University Press |year=2001}}</ref>

Ultimately, as a consequence of the isoperimetric inequality, {{pi}} appears in the optimal constant for the critical [[Sobolev inequality]] in ''n'' dimensions, which thus characterizes the role of {{pi}} in many physical phenomena as well, for example those of classical [[potential theory]].<ref>{{cite journal |last=Talenti |first=Giorgio |title=Best constant in Sobolev inequality |journal=Annali di Matematica Pura ed Applicata |volume=110 |number=1 |pages=353–372 |issn=1618-1891 |doi=10.1007/BF02418013 |citeseerx=10.1.1.615.4193 |year=1976 |s2cid=16923822}}</ref><ref>{{cite arXiv |title=Best constants in Poincaré inequalities for convex domains |eprint=1110.2960 |author1=L. Esposito |author2=C. Nitsch |author3=C. Trombetti |year=2011 |class=math.AP}}</ref><ref>{{cite journal |title=Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions |first1=M. |last1=Del Pino |first2=J. |last2=Dolbeault |journal=Journal de Mathématiques Pures et Appliquées |year=2002 |volume=81 |issue=9 |pages=847–875 |doi=10.1016/s0021-7824(02)01266-7 |citeseerx=10.1.1.57.7077 |s2cid=8409465}}</ref>  In two dimensions, the critical Sobolev inequality is

<math display=block>2\pi\|f\|_2 \le \|\nabla f\|_1</math>

for ''f'' a smooth function with compact support in {{math|'''R'''<sup>2</sup>}}, <math>\nabla f</math> is the [[gradient]] of ''f'', and <math>\|f\|_2</math> and <math>\|\nabla f\|_1</math> refer respectively to the [[Lp space|{{math|L<sup>2</sup>}} and {{math|L<sup>1</sup>}}-norm]].  The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.

Wirtinger's inequality also generalizes to higher-dimensional [[Poincaré inequality|Poincaré inequalities]] that provide best constants for the [[Dirichlet energy]] of an ''n''-dimensional membrane.  Specifically, {{pi}} is the greatest constant such that

<math display=block>
\pi \le \frac{\left (\int_G |\nabla u|^2\right)^{1/2}}{\left (\int_G|u|^2\right)^{1/2}}
</math>

for all [[convex set|convex]] subsets {{math|''G''}} of {{math|'''R'''<sup>''n''</sup>}} of diameter 1, and square-integrable functions ''u'' on {{math|''G''}} of mean zero.<ref>{{cite journal |last1=Payne |first1=L. E. |last2=Weinberger |first2=H. F. |title=An optimal Poincaré inequality for convex domains |year=1960 |journal=Archive for Rational Mechanics and Analysis |volume=5 |issue=1 |issn=0003-9527 |pages=286–292 |doi=10.1007/BF00252910 |bibcode=1960ArRMA...5..286P |s2cid=121881343}}</ref>  Just as Wirtinger's inequality is the [[calculus of variations|variational]] form of the [[Dirichlet eigenvalue]] problem in one dimension, the Poincaré inequality is the variational form of the [[Neumann problem|Neumann]] eigenvalue problem, in any dimension.