In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle. It is named after the French mathematician Adhémar Jean Claude Barré de Saint-Venant. Given a simply connected domain D in the plane with area A, ρ {\displaystyle \rho } the radius of its greatest inscribed circle, the torsional rigidity P of D is defined by
P = 4 sup f ( ∬ D f d x d y ) 2 ∬ D f x 2 + f y 2 d x d y . {\displaystyle P=4\sup _{f}{\frac {\left(\iint \limits _{D}f\,dx\,dy\right)^{2}}{\iint \limits _{D}{f_{x}}^{2}+{f_{y}}^{2}\,dx\,dy}}.}
Here the supremum is taken over all the continuously differentiable functions vanishing on the boundary of D. The existence of this supremum is a consequence of Poincaré inequality. Saint-Venant conjectured in 1856 that of all domains D of equal area A the circular one has the greatest torsional rigidity, that is
P ≤ P circle ≤ A 2 2 π . {\displaystyle P\leq P_{\text{circle}}\leq {\frac {A^{2}}{2\pi }}.}
A rigorous proof of this inequality was not given until 1948 by Pólya. Another proof was given by Davenport and reported in. A more general proof and an estimate
P < 4 ρ 2 A {\displaystyle P<4\rho ^{2}A}
is given by Makai.
Notes