In measure theory, a branch of mathematics, the ham sandwich theorem, also called the Stone–Tukey theorem after Arthur H. Stone and John Tukey, states that given n' measurable "objects" in n-dimensional space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single (n − 1)-dimensional hyperplane. Here the "objects" should be sets of finite measure (or, in fact, just of finite outer measure) for the notion of "dividing the volume in half" to make sense.
The ham sandwich theorem takes its name from the case when n = 3 and the three objects of any shape are a chunk of ham and two chunks of bread — notionally, a sandwich — which can then all be simultaneously bisected with a single cut (i.e., a plane). In two dimensions, the theorem is known as the pancake theorem of having to cut two infinitesimally thin pancakes on a plate each in half with a single cut (i.e., a straight line).
The ham sandwich theorem is also sometimes referred to as the "ham and cheese sandwich theorem", again referring to the special case when n = 3 and the three objects are
a chunk of ham,
a slice of cheese, and
two slices of bread (treated as a single disconnected object).
The theorem then states that it is possible to slice the ham and cheese sandwich in half such that each half contains the same amount of bread, cheese, and ham. It is possible to treat the two slices of bread as a single object, because the theorem only requires that the portion on each side of the plane vary continuously as the plane moves through 3-space.
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