Gruber P. Convex and Discrete Geometry

180 Convex Bodies
Torsional Rigidity
Let K be a smooth closed Jordan curve in E2 . It is the boundary of a compact body
B, where by a body we mean a compact set which equals the closure of its interior.
Consider a cylindrical rod of homogeneous elastic material with cross-section B.
The torsional rigidity T (B) of the rod is the torque required for a unit angle of twist
per unit length under the assumption that the shear modulus is 1. It can be expressed
in the form
�
� �2dx, (1) T (B) = grad u(x)
B
where u : B → R is the solution of the following boundary value problem, where
is the Laplace operator:
(2) �u + 2 = 0inintB
u|K = 0
u|B is continuous on B
u| int B is of class C 2
� = ∂2
∂x 2 1
+ ∂2
∂x 2 2
The following result was conjectured by Saint-Venant [875] and first proved by Pólya
[809] a century later.
Theorem 9.8. Let K be a closed Jordan curve of class C 1 and B the compact body
in E 2 bounded by K and of area A(B). Then
T (B) ≤ A(B)2
2π .
Equality is attained if B is a circular disc.
Proof. First, a different representation of T (B) will be given:
⎧ � �
⎪⎨ 2 f (x) dx
B
(3) T (B) = sup
⎪⎩
�2 � � �2dx :
grad f (x)
B
⎫
⎬
f : B →[0, +∞) locally Lipschitz, f �= 0, f |K = 0
⎭ ,
where the supremum is attained precisely for f of the form f = u �= 0, u
as in (2).
Let ∂
∂n denote the derivative in the direction of the exterior unit normal vector of K
and let x(·) :[0, L] →E2 be a parametrization of K where the arc-length s is the
parameter and L the length of K . Recall the following formula of Green for integrals.