<Dd> π 2 ∫ 0 1 f (x) 2 d x ≤ ∫ 0 1 f ′ (x) 2 d x, (\ displaystyle \ pi ^ (2) \ int _ (0) ^ (1) f (x) ^ (2) \, dx \ leq \ int _ (0) ^ (1) f' (x) ^ (2) \, dx,) </Dd> <P> and the case of equality holds precisely when f is a multiple of sin (π x). So π appears as an optimal constant in Wirtinger's inequality, and from this it follows that it is the smallest such wavenumber, using the variational characterization of the eigenvalue . As a consequence, π is the smallest singular value of the derivative on the space of functions on (0, 1) vanishing at both endpoints (the Sobolev space H 0 1 (0, 1) (\ displaystyle H_ (0) ^ (1) (0, 1))). </P> <P> 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 </P> <Dl> <Dd> 4 π A ≤ P 2, (\ displaystyle 4 \ pi A \ leq P ^ (2),) </Dd> </Dl>

21 is the geometric mean of 7 and what other number