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For a measure on a subset of the complex plane we consider $L^p$-optimalweighted polynomials, namely, monic polynomials of degree $n$ with a varyingweight 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 boundednumber of the roots of the $L^p$-optimal polynomials lie within a smallneighborhood of the support of a certain equilibrium measure; asymptotics forthe $n$th roots of the $L^p$ norms are also provided. The case $p=\infty$ iswell 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.

On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights