Pizza theorem with realistic extension

Question

As I was cutting up a pizza today, fresh out of the oven, I thought of the pizza theorem again (this happens pretty often). As you may know, the pizza theorem provides a guaranteed way to divide a pizza into equal areas using only cuts that form chords on the pizza circle, rotated about a point located somewhere on the pizza.

Then I got to thinking, though. What about the crust? Everyone wants an equal share of that also, right? (Excluding those barbarians who leave their crusts on the plate. Barbarians.)

I don't have a proof, but it seems to me that if the crust is of uniform width, the regular pizza theorem will still work (for the same reason that the radius of the pizza, if constant, does not affect the algorithm). On the other hand, crusts can get pretty wavy, sometimes. Additionally, the larger crust takes up area that would be otherwise occupied by pizza proper!

So, let's say we have a continuous, always differentiable function $w(\theta)$ defined on the interval $[0,2\pi)$ (and repeated periodically thereafter) that specifies the width of the crust at any point. The radius of the remaining pizza $r(\theta)$ at any given angle, then, is $r_0 - w(\theta)$, where $r_0$ is the constant radius of the whole pizza.

I think that there must still exist a solution that provides equal crust and equal pizza center for any $n/4$ eaters and $n$ cuts, because the pizza can be rotated freely until the first cut. Is that so? How do you do it? (It may be that the factor is no longer $1/4$.)

Answer 1

It is always important to know what the theorem says, before attempting to apply it.

The pizza theorem says that if you cut a pizza into 4n slices at equal angles, then you can divide it equally amongst n people by giving them slices in order, regardless of the point of intersection, or the angle of first cut.

To prove that the uniform crust scenario leads to equal crusts, you just apply the pizza theorem twice. Once to the entire pizza, and once to the stuffing. These can be equally distributed, and hence their difference is equal.

For the case where crust is not uniform, the restriction of equal angles easily allows you to construct a scenario which violates equal crust. For example, when the crust is consolidated at one point, and so at most 2 people can have crust.

If however, you relax the requirement that the slices occur at equal angles, and you have a random starting point, then it's a whole different ball game.