  It has recently been established byWang and Xia [WX] that local minimizers of
perimeter within a ball subject to a volume constraint must be spherical caps
or planes through the origin. This verifies a conjecture of the authors and is
in contrast to the situation of area minimizing surfaces with prescribed
boundary where singularities can be present in high dimensions. This result
lends support to the more general conjecture that volume-constrained minimizers
in arbitrary convex sets may enjoy better regularity properties than their
boundary-prescribed cousins. Here, we show the importance of the convexity
condition by exhibiting a simple example, given by the Simons cone, of a
singular volume-constrained locally area-minimizing surface within a nonconvex
domain that is arbitrarily close to the unit ball.
