Can a positively charged conductor have negative surface charge density somewhere? 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.
 A: Our body is positively charged, so it's potential is positive.
Suppose there is a place on it's surface with a negative charge density. Electric field direction is towards the surface near the spot.
Consider some electric field line that ends on our spot. Where could it originate?
There are only two options: it originated either from some other point of the conductor or from the infinity. (*)
The first option would mean that two points of the conductor have different potentials, so it's impossible.
Second option would mean that the potential of ending point of the electric line field is negative. But it must be positive.
So there are no negatively charged regions on the surface of our body.
(*) I understand that the proof is not rigorous here, but I am quite sure I can improve it.
A: I am not sure I fully understand the question, but it seems to me that it is "possible for some region to end up with negative charge even if the entire conductor has positive charge".
Let us place a perfectly conducting cone and a perfectly conducting sphere of a large radius in such a way that the cone and the sphere do not overlap, the center of the sphere is on the extension of the axis of the cone, and the tip of the cone is close to the surface of the sphere. Let us connect the sphere and the cone by a conductor that is placed far from the tip of the cone and put positive charge on the resulting conducting system including the cone, the sphere, and the conductor. I would think that there will be high electric field near the tip of the cone that would repel positive charges on the sphere in the vicinity of the tip, so there will probably be a negative charge density on the sphere near the edge of the cone.
A: This is not possible. The potential on the surface of the conductor has the same positive value everywhere. For a negative surface charge to exist, a field line should connect to a location with higher potential and this does not exist. Only locations with lower potential exist, namely locations that are connected to the conductor and to infinity, where the potential is zero. 
A: Yes, of course a positively charged object can have a
negatively charged surface.   Consider a hollow sphere
of metal,charged A,  with an insulated positive-charge, B, inside the
hollow.   The exterior surface of the sphere will have
the sum (A + B) charge, but the interior
surface of the sphere will hold an induced (-B)
charge.
