Making a cut trough a center of mass, can the masses of the pieces be equal? Let's say point $P$ is the center of mass of an irregularly shaped object. 


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*If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same mass? This would be possible if the centers of mass of each piece was the same distance from $P$, but is this possible with an irregularly shaped object of uniform density? 

*What about non-uniform density? 
This is just a question I thought about.
 A: You can do even better.  The ham sandwich theorem says you can simultaneously bisect any three volumes in space with a single plane.  You can have a plane through the center of mass that bisects the mass and the surface area.  Make one volume a tiny sphere around the center of mass, one a very thin outer skin, and the other the rest of the object.  The proof looks to me like it applies to non-uniform density as well.
A: Yes, this is always possible for any object of any shape.
I will explain why this is the case:
Take any object and make any cut through the center of mass.
The object is now divided in two parts, A and B.
Let's say A has a higher mass than B.
We can now start rotating the plane of the cut until it has made a half rotation.
After this half rotation the masses of A and B have turned around, so now mass B is greater than mass A.
During this turning process the mass of A has been decreasing and the mass of B has been increasing. And since both are continuous at some point their masses must have been equal.
Even with a non-uniform density, the lines remain continuous and so they must still intersect at some point. 
Thus, for any point and any object with any denisity, there is some cut through that point that divides the object in two pieces of equal mass.
