It reminds me of a couple of other interesting things from analysis, but I can’t recall sufficient detail at this time …

Anyway, your mention of mean and balance point reminded me of a prop I used to introduce the mean in stat class, which always reminds me of one of my favorite demonstrations of introductory physics, elegantly demonstrating dynamic friction : Take a walking stick, or some other stick whose center of mass is significantly away from the center of the longest dimension, preferably about 1/3 from one end or so. Hold the stick up supported by your two extended index fingers, with one finger on or near each end, but such that the center of mass is significantly far away from both points of contact. It is easiest to extend both index fingers as if you were pointing with them, and to rest the stick on them. It is then making contact with the outer sides of the fingers, now facing upwards. Then smoothly draw both fingers together. If done carefully your fingers will intersect directly below the center of mass, leaving the stick conspicuously balancing on a small area. This is not very difficult to achieve, and can be done with about any stick or dowel I’ve tried, including swords, clubs, bats, etc. The center of mass does not need to be anywhere special, but if it is about 1/3 from one end the demonstration looks particularly nice IMO.

A physical analogy is that the center of mass looks at torque and the ham sandwich theorem looks just at mass. There could be more mass on one side of the balance point than the other.

In terms of probability, the center of mass is like the mean, but the ham sandwich theorem is looking at medians.

I guess I’m confused about this fact, though — I thought the definition of the center of mass implied that any plane containing it would bisect the shape perfectly (assuming unform mass density). Surely this result does not rely on inhomogenous mass density, though, or it would be trivial to obtain any ratio one pleases.

Like Scott, I’d sure like to see a counterexample!