The ham sandwich theorem states that given objects floating in -dimensional space, there exists a single -dimensional hyperplane that simultaneously cuts all objects into two pieces of equal volume. In the case , this theorem states that if two pieces of bread and one piece of ham are floating in three-dimensional space, then there is a single plane that slices each of the three items into two equal-volume chunks (hence, the name "ham sandwich theorem").
In the case , the ham sandwich theorem states that given two disjoint regions of the plane, there is a line that simultaneously divides both regions into two pieces of equal area. This special case is known as the pancake theorem, since regions of the plane can look a bit like pancakes.
Let be pancakes, with . There exists a line that simultaneously slices both and into pieces of equal area.
The pancake theorem can be proved using the intermediate value theorem:
Each direction in the plane has a corresponding unit vector, which can be thought of as a point in the circle . For each direction , there is a unique line with slope that divides the region into two pieces of equal area.
Furthermore, the line divides into two regions, which we arbitrarily denote the positive side and the negative side . Define a function by That is, equals the area of the part of located on the positive side of . Since is a continuous function, we can apply the -dimensional Borsuk-Ulam theorem. This implies there is some such that .
Note that is precisely , since the positive side of is precisely the negative side of . Thus, . This means bisects . But we constructed so that bisects for all . Thus, the line simultaneously bisects and !
The proof of the ham sandwich theorem for is essentially the same but requires a higher-dimensional analog of the Borsuk-Ulam theorem. Unfortunately, the intermediate value theorem does not suffice to prove these higher-dimensional analogs; one needs to use the machinery of algebraic topology.