This is a simple question that occurred to me while thinking about electrostatics. Let's consider a positively charged isolated conductor in equilibrium. In general, the surface charge density varies over the surface, peaking at regions of sharp curvature. This is fairly intuitive, but it becomes less intuitive when we allow the conductor to be concave, with any possible shape for its perimeter.
If we allow an arbitrary (connected) shape, is it possible for some region to end up with negative charge even if the entire conductor has positive charge? I suspect not, but I can't prove it can't happen.