Here’s a surprising theorem. Suppose you have a convex set in Rn you pass a plane through its center of mass. How much of the volume of the set can be on one side? No more than about 63%. (Precisely, 1 – 1/e.) This holds for any dimension n and for any direction through the center. I don’t have a reference for this theorem except that it is mentioned near the end of lecture 5 in this course.
Update: See Splitting a convex set through its center for an illustration and a partial proof.
I am curious what shape the limit set has. Can you actually attain 1 – 1/e? What does such a set look like for n=2 and n=3?
The limit depends on dimension. When n = 1, convex sets are intervals, and so all convex sets split perfectly at their center. I believe the ratio increases monotonically as a function of n and quickly approaches 1 – 1/e. I think the result was proved in the 1960’s. But this is all second hand. I’m just going by what I heard in the lecture. I haven’t seen the details.
Ok, that makes sense. But I still wonder what this looks like in dimensions 2 and 3. What is the maximum area for n=2 and the maximum volume for n=3? And what exactly are these maximally lopsided shapes?
I don’t know about the lopsided shapes, but I do recall proving the “ham sandwich theorem” in a math final.
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!
John,
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.
Aha. I understand now. Interesting! It certainly is plausible and I can see why restricing it to convex sets makes it an interesting problem.
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 follow up post answers some of the questions raised here.
@John
Is it (1 + 1/n)^n for R^n ?
ooops I mean 1 – (1 + 1/n)^(-n)