The Ham and Cheese Sandwich Theorem

My patient plodding through the definitions and basic theorems needed to get Topology underway has at last been rewarded with substantial result. The formal statement is:

Given three finite volumes, U, V, and W in Cartesian 3-space, there is at least one plane that bisects all three.

The application to the bisection of a ham and cheese sandwich with one cut should be obvious. Note that no assumption is made about the way the ham and the cheese are distributed within the sandwich.

The proof is quite short and fairly easy to understand, though it does depend on the Borsuk-Ulam Theorem, which my book considers too hard to prove except in one special case.