Sharing a pizza: bisecting masses with two cuts

04/04/2019

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by   Luis Barba, et al.

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Assume you have a pizza consisting of four ingredients (e.g., bread, tomatoes, cheese and olives) that you want to share with your friend. You want to do this fairly, meaning that you and your friend should get the same amount of each ingredient. How many times do you need to cut the pizza so that this is possible? We will show that two straight cuts always suffice. More formally, we will show the following extension of the well-known Ham-sandwich theorem: Given four mass distributions in the plane, they can be simultaneously bisected with two lines. That is, there exist two oriented lines with the following property: let R^+_1 be the region of the plane that lies to the positive side of both lines and let R^+_2 be the region of the plane that lies to the negative side of both lines. Then R^+=R^+_1∪ R^+_2 contains exactly half of each mass distribution.