Ham Sandwich Theorem

Here is one of my favorite theorems from topology, called the Ham Sandwich Theorem.

It says: given globs of ham, bread, and cheese (in any shape), placed any way you like, there exists one flat slice of a knife (a plane) that will bisect each of the ham, bread, and cheese.

In other words you can share it with a friend so that both of you get exactly the same amounts of all three globs!

Moreover, the ham, bread, and cheese don’t even need to be near each other, nor do they have to be connected!

Presentation Suggestions:
Get your class to envision various scenarios for the location of the ham, bread, and cheese, and see if they can figure out where the plane should go. By the way, a common error is to think that the plane should pass through the centers of mass of the three objects. But this is not necessarily the case, as the reader can check by constructing some simple examples using lopsided volumes.

The Math Behind the Fact:
There is a version that holds in N-dimensional space, that says any N globs of positive volume can be simultaneously bisected by a single hyperplane. Like the Brouwer fixed point theorem and the Borsuk-Ulam theorem, this has an existence proof… it doesn’t say where the plane is! Actually, the Ham Sandwich Theorem can be proved using the Borsuk-Ulam theorem.

How to Cite this Page: 
Su, Francis E., et al. “Ham Sandwich Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Francis Su

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