  Let $\Gamma \subset \mathbb C$ be a curve of class $C(2,\alpha)$. For $z_{0}$
in the unbounded component of ${\mathbb C}\setminus \Gamma$, and for
$n=1,2,...$, let $\nu_n$ be a probability measure with supp$(\nu_{n})\subset
\Gamma$ which minimizes the Bergman function
$B_{n}(\nu,z):=\sum_{k=0}^{n}|q_{k}^{\nu}(z)|^{2}$ at $z_{0}$ among all
probability measures $\nu$ on $\Gamma$ (here,
$\{q_{0}^{\nu},\ldots,q_{n}^{\nu}\}$ are an orthonormal basis in $L^2(\nu)$ for
the holomorphic polynomials of degree at most $n$). We show that
$\{\nu_{n}\}_n$ tends weak-* to $\hat\delta_{z_{0}}$, the balayage of the point
mass at $z_0$ onto $\Gamma$, by relating this to an optimization problem for
probability measures on the unit circle. Our proof makes use of estimates for
Faber polynomials associated to $\Gamma$.
