Two friends wants to share a pizza. One of them loves the edge of the pizza and the other one hates it. Both consider the pizza to get tastier the closer to the center you get. What is the fairest way to cut the pizza if you are only allowed four straight cuts, such that the two sets of slices sum up to the same area and one set contains all edges?
I have attached an intuitive sketch, which seems somewhat fair, but I don't know how to approach a problem like this one.
On
Since I raised the possibility in one of my comments, here is a possible solution that gives ~42% of the center to the edge lover. It does however rely on the freedom to displace the pieces between the cuts. Given that this is a problem of practical nature, I don't see any harm in allowing displacement. Folding the pizza before or between cuts would probably result in undesired side effects, but it is possible that could yield an even fairer distribution.
About all you can do is to show that a solution exists under certain conditions using the intermediate value function. If you are given a specific function for their valuation of the various areas of the pizza you may be able to derive a specific distribution. As long as the valuation is not too strongly peaked at the center you can note that having four points near each other on the edge gives the one who does not like crust almost nothing. Having four points that form a square gives the one who does not like crust $\frac \pi 4$ of the area including the desirable center. Somewhere in between is a set of vertices that gives them each the same valuation.
If $90\%$ of the value of the pizza is the center point and the rest is evenly spread, you can't be fair. You have to make one segment go through the center, but there is not enough freedom to split the rest evenly with only four cuts.