OK, I think my problem is that I never internalized the fact that the calculation of center of mass calculation is weighted by the relative moment of each part of the mass. I don’t know how I got through a math degree without really being cognizant of this.
So I guess my question is: does an arbitrary volume contain a uniquely defined (point) center such that any dividing hyperplane through that point bisects the volume into two equal volumes? I had always assumed yes, and that it would be easy to find, but I guess maybe not. And now that I think about it there are plenty of trivial counterexamples (e.g., a three-pointed star shape).
