On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint

We consider the problem of locally minimizing perimeter within a given boun ded domain Omega subset of R-n subject to a volume constraint. By a local m inimizer, we mean a set of finite perimeter E subset of Omega satisfying th e condition P(E, Omega) less than or equal to P(F, Omega) among all competitors F subset of Omega such that \F\ = \E\ and such that p arallel to chi(F) - chi(E)parallel to(L1(Omega)), < delta for some delta > 0. We prove that when Omega is convex, the boundary partial derivative E bo olean AND Omega is connected, or else partial derivative E boolean AND Omeg a consists of parallel planes meeting partial derivative Omega orthogonally . The result arises as an application of a property we derive for normal va riations of constant mean curvature hypersurfaces bounding sets within a co nvex domain Omega. The property states that for such variations, area is a concave function of the enclosed volume. Our results hold in all dimensions n, even in the presence of singularities.