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
Saint-Venant conjectured in 1856 that of all domains D of equal area A the circular one has the greatest torsional rigidity, that is
is given by Makai.
- E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966doi:10.1007/BF01894885
- A J-C Barre de Saint-Venant, Mémoire sur la torsion des prismes, Mémoires présentés par divers savants à l'Académie des Sciences, 14 (1856), pp. 233–560.
- G. Pólya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267, 277.
- G. Pólya and G. Szegő, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).