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, the radius and the area of its greatest inscribed circle, the torsional rigidity P
of D is defined by
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
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
is given by Makai.