Question about distribution of mass

Question

I recently began taking my first university-level physics course after having studied quite a bit of pure mathematics. While I think that my math background has helped me grasp some concepts a bit more quickly, there are still some topics I can accept intellectually but which I need some help wrapping my head around on a more intuitive level. One of these topics is center of mass. When I was first introduced to the concept, I (incorrectly) thought that the center of mass is defined as the point on an object about which mass is equally distributed. While I can accept that the center of mass is in fact simply the average location of mass, and might more accurately be termed the "center of torque," I have been trying to understand my original, erroneous definition of the center of mass more deeply to try to fix my own misunderstandings. I have begun to wonder whether a point like the one I initially described can be guaranteed to exist in every object at all. As such, I pose this question: let S be a 2-dimensional object whose total mass is M and whose mass at a given point (x,y) on S is given by the function M'(x). Moreover, assume that while density is not necessarily constant across various regions of S, M' is continuous on S. Does there exist a point P such that, given any line L through P, L bisects the mass of S? Note that P need not be on S. If P does exist, is the a formula to derive it given the mass and location of each point on S? Moreover, does density need be continuous for the existence of P to be guaranteed, and do analogs of P exist in higher dimensions? If P does not always exist, could it shown that P always exists if we are allowed to assume that density is constant?

Answer 1

Does there exist a point P such that, given any line L through P, L bisects the mass of S?

Not necessarily. Counterexample - place three equal point masses $m$ at three corners of an equilateral triangle, connected by light rods (so light that we can ignore the masses of the rods). If P exists, it must lie on some line that passes through a corner of the triangle, so it has a mass $m$ on each side of it. But any line through P that does not pass through a corner of the triangle has mass $2m$ on one side of it and only $m$ on the other side. So wherever we place P, there are many lines through P that do not bisect the masses on the triangle.