  Is the closest product state to a symmetric entangled multiparticle state
also symmetric? This question has appeared in the recent literature concerning
the geometric measure of entanglement. First, we show that a positive answer
can be derived from results concerning symmetric multilinear forms and
homogeneous polynomials, implying that the closest product state can be chosen
to be symmetric. We then prove the stronger result that the closest product
state to any symmetric multiparticle quantum state is necessarily symmetric.
Moreover, we discuss generalizations of our result and the case of
translationally invariant states, which can occur in spin models.
