  In this note we address the attempted proof of the existence of static
solutions to the Einstein-Vlasov system as given in \cite{Wol}. We focus on a
specific and central part of the proof which concerns a variational problem
with an obstacle. We show that two important claims in \cite{Wol} are incorrect
and we question the validity of a third claim. We also discuss the variational
problem and its difficulties with the aim to stimulate further investigations
of this intriguing problem: to answer the question whether or not static
solutions of the Einstein-Vlasov system can be found as local minimizers of an
energy-Casimir functional.
