  Let $E$ be a compact set of positive logarithmic capacity in the complex
plane and let $\{P_n(z)\}_{1}^{\infty}$ be a sequence of asymptotically
extremal monic polynomials for $E$ in the sense that
\begin{equation*}%\label{}
\limsup_{n\to\infty}\|P_n\|_E^{1/n}\le\mathrm{cap}(E). \end{equation*} The
purpose of this note is to provide sufficient geometric conditions on $E$ under
which the (full) sequence of normalized counting measures of the zeros of
$\{P_n\}$ converges in the weak-star topology to the equilibrium measure on
$E$, as $n\to\infty.$ Utilizing an argument of Gardiner and Pommerenke dealing
with the balayage of measures, we show that this is true, for example, if the
interior of the polynomial convex hull of $E$ has a single component and the
boundary of this component has an "inward corner" (more generally, a
"non-convex singularity"). This simple fact has thus far not been sufficiently
emphasized in the literature. As applications we mention improvements of some
known results on the distribution of zeros of some special polynomial
sequences.
