Saint-Venant's theorem

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, $\rho$ the radius and $\sigma$ the area of its greatest inscribed circle, the torsional rigidity P of D is defined by

$P=4\sup _{f}{\frac {\left(\iint \limits _{D}f\,dx\,dy\right)^{2}}{\iint \limits _{D}{f_{x}}^{2}+{f_{y}}^{2}\,dx\,dy}}.$

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

$P\leq P_{\text{circle}}\leq {\frac {A^{2}}{2\pi }}.$

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

$P<4\rho ^{2}A$

is given by Makai.