We consider the problem of locally minimizing perimeter within a given bounded domain fi C M n subject to a volume constraint. By a local minimizer, we mean a set of finite perimeter E C O satisfying the condition P(E,ft)<P(F,ft) among all competitors F C Q such that \ = \ and such that ||XF -X^IILUQ) < ^ f or some 6 > 0. We prove that when Q is convex, the boundary dE Pi O is connected, or else dE Pi O consists of parallel planes meeting 9fi orthogonally. The result arises as an application of a property we derive for normal variations of constant mean curvature hypersurfaces bounding sets within a convex domain Q. 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.
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Sternberg et al. (1999) studied this question.