section_path: Role and characterizations in mathematics/Inequalities
section_name: Inequalities
paragraph_text: Inequalities
  
  The number π serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentioned above, it can be characterized via its role as the best constant in the isoperimetric inequality: the area A enclosed by a plane Jordan curve of perimeter P satisfies the inequality 4 π A ≤ P 2, {\displaystyle 4\pi A\leq P^{2},} and equality is clearly achieved for the circle, since in that case A = π r and P = 2π r.
  
  Ultimately, as a consequence of the isoperimetric inequality, π appears in the optimal constant for the critical Sobolev inequality in n dimensions, which thus characterizes the role of π in many physical phenomena as well, for example those of classical potential theory. In two dimensions, the critical Sobolev inequality is 2 π ‖ f ‖ 2 ≤ ‖ ∇ f ‖ 1 {\displaystyle 2\pi \|f\|_{2}\leq \|\nabla f\|_{1}} for f a smooth function with compact support in R, ∇ f {\displaystyle \nabla f} is the gradient of f, and ‖ f ‖ 2 {\displaystyle \|f\|_{2}} and ‖ ∇ f ‖ 1 {\displaystyle \|\nabla f\|_{1}} refer respectively to the L and L -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 inequalities that provide best constants for the Dirichlet energy of an n -dimensional membrane. Specifically, π is the greatest constant such that π ≤ (∫ G | ∇ u | 2) 1 / 2 (∫ G | u | 2) 1 / 2 {\displaystyle \pi \leq {\frac {\left(\int _{G}|\nabla u|^{2}\right)^{1/2}}{\left(\int _{G}|u|^{2}\right)^{1/2}}}} for all convex subsets G of R of diameter 1, and square-integrable functions u on G of mean zero. Just as Wirtinger's inequality is the variational form of the Dirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of the Neumann eigenvalue problem, in any dimension.