  We prove that every hyperplane passing through the origin in $\rr^{n+1}$
divides an embedded compact free boundary minimal hypersurface of the euclidean
$(n+1)$-ball in exactly two connected hypersurfaces. We also show that if a
region in the $(n+1)$-ball has mean convex boundary and contains a
nullhomologous $(n-1)$-dimensional equatorial disk, then this region is a
closed halfball. Our first result gives evidence to a conjecture by Fraser and
Li in any dimension.
