  A mean-convex set can be regarded as a barrier for the construction of
minimal surfaces. Namely, if we are given a mean-convex set and a
null-homotopic Jordan curve on its boundary, then there exists an embedded
minimal disk with boundary the given curve contained in the starting
mean-convex set. Does a mean-convex set contain all minimal disks with boundary
on it? Does it contain the solutions of Plateau's problem? We answer this
question negatively and characterize the least barrier enclosing all the
minimal hypersurfaces with boundary on a given set.
