  Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $u$ be
the solution of $-\Delta u = 1$ vanishing on the boundary $\partial \Omega$.
The estimate $$ \| \nabla u\|_{L^{\infty}(\Omega)} \leq c |\Omega|^{1/2}$$ is
classical. We use the P-functional, the stability theory of the torsion
function and Brownian motion to establish the estimate for a universal $c <
(2\pi)^{-1/2}$. We also give a numerical construction showing that the optimal
constant satisfies $c \geq 0.358$. The problem is important in different
settings: (1) as the maximum shear stress in Saint Venant Elasticity Theory,
(2) as an optimal control problem for the constrained maximization of the
lifetime of Brownian motion started close to the boundary and (3) and optimal
Hermite-Hadamard inequalities for subharmonic functions on convex domains.
