  For a measure on a subset of the complex plane we consider $L^p$-optimal
weighted polynomials, namely, monic polynomials of degree $n$ with a varying
weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1
\leq p \leq \infty$. It is shown that eventually all but a uniformly bounded
number of the roots of the $L^p$-optimal polynomials lie within a small
neighborhood of the support of a certain equilibrium measure; asymptotics for
the $n$th roots of the $L^p$ norms are also provided. The case $p=\infty$ is
well known and corresponds to weighted Chebyshev polynomials; the case $p=2$
corresponding to orthogonal polynomials as well as any other $1\leq p <\infty$
is our contribution.
