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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.^{[1]} 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^{[2]} 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.^{[3]} Another proof was given by Davenport and reported in.^{[4]} A more general proof and an estimate

is given by Makai.^{[1]}

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^{a}^{b}E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966 doi:10.1007/BF01894885 **^**A J-C Barre de Saint-Venant,popularly known as संत वनंत 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).