Area minimizing sets subject to a volume constraint in a convex set

For a given convex subset Ω of Euclidean n-space, we consider the problem of minimizing the perimeter of subsets of Ω subject to a volume constraint. The problem is to determine whether in general a minimizer is also convex. Although this problem is unresolved, we show that if Ω satisfies a “great circle” condition, then any minimizer is convex. We say that Ω satisfies a great circle condition if the largest closed ball B contained in Ω has a great circle that is contained in the boundary of Ω. A great circle of B is defined as the intersection of the boundary of B with a hyperplane passing through the center of B.