Ham Sandwich Theorem

The ham sandwich theorem states that given \(n\) objects floating in \(n\)-dimensional space, there exists a single \((n-1)\)-dimensional hyperplane that simultaneously cuts all \(n\) objects into two pieces of equal volume. In the case \(n=3\), 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").

This is a ham sandwich.

In the case \(n=2\), 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.

There is a line that simultaneously bisects both blue pancakes.

Let \(K_1, K_2 \subset \mathbb{R}^2\) be pancakes, with \(K_1 \cap K_2 = \emptyset\). There exists a line that simultaneously slices both \(K_1\) and \(K_2\) into pieces of equal area.

The pancake theorem can be proved using the intermediate value theorem:

Each direction in the plane \(\mathbb{R}^2\) has a corresponding unit vector, which can be thought of as a point in the circle \(S^1 = \big\{(x,y) \in \mathbb{R}^2 \, : \, x^2 + y^2 = 1\big\}\). For each direction \(m = \big(\cos(\theta), \sin(\theta)\big) \in S^1\), there is a unique line \(\ell (m)\) with slope \(\tan(\theta)\) that divides the region \(K_1\) into two pieces of equal area.

Furthermore, the line \(\ell (m)\) divides \(\mathbb{R}^2\) into two regions, which we arbitrarily denote the positive side \(P (m)\) and the negative side \(N (m)\). Define a function \(f: S^1 \to \mathbb{R}\) by \[ f(m) := \text{Area} \big(K_2 \cap P (m)\big).\] That is, \(f(m)\) equals the area of the part of \(K_2\) located on the positive side of \(\ell(m)\). Since \(f\) is a continuous function, we can apply the \(1\)-dimensional Borsuk-Ulam theorem. This implies there is some \(n\in S^1\) such that \(f(n) = f(-n)\).

Note that \(f(-n)\) is precisely \(\text{Area} \big(K_2 \cap N (n)\big)\), since the positive side of \(\ell(-n)\) is precisely the negative side of \(\ell(n)\). Thus, \(\text{Area}\big(K_2 \cap P(n)\big) = \text{Area}\big(K_2 \cap N(n)\big)\). This means \(\ell(n)\) bisects \(K_2\). But we constructed \(\ell\) so that \(\ell(m)\) bisects \(K_1\) for all \(m \in S^1\). Thus, the line \(\ell(n)\) simultaneously bisects \(K_1\) and \(K_2\)! \(_\square\)

The proof of the ham sandwich theorem for \(n>2\) 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.