A two-piece property for free boundary minimal hypersurfaces in the (n + 1)-dimensional ball

We prove that every hyperplane passing through the origin in Rn+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.