  We consider the problem of finding the best approximation to $\bar{z}$ in the
Bergman Space $A^{2}(\Omega)$. We show that this best approximation is the
derivative of the solution to the Dirichlet problem on $\partial\Omega$ with
data $\left|z\right|^{2}$ and give examples of domains where the best
approximation is a polynomial, or a rational function. Finally, we obtain the
"isoperimetric sandwich" for $dist(\overline{z},\Omega)$ that yields the
celebrated St. Venant inequality for torsional rigidity.
