Ham sandwich theorem

In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable 'objects' in n-dimensional Euclidean 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. In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable 'objects' in n-dimensional Euclidean 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. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without bothering to automatically state the theorem in the n-dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey. 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 because of having to cut two infinitesimally thin pancakes on a plate each in half with a single cut (i.e., a straight line). According to Beyer & Zardecki (2004), the earliest known paper about the ham sandwich theorem, specifically the n = 3 case of bisecting three solids with a plane, is by Steinhaus (1938). Beyer and Zardecki's paper includes a translation of the 1938 paper. It attributes the posing of the problem to Hugo Steinhaus, and credits Stefan Banach as the first to solve the problem, by a reduction to the Borsuk–Ulam theorem. The paper poses the problem in two ways: first, formally, as 'Is it always possible to bisect three solids, arbitrarily located, with the aid of an appropriate plane?' and second, informally, as 'Can we place a piece of ham under a meat cutter so that meat, bone, and fat are cut in halves?' Later, the paper offers a proof of the theorem. A more modern reference is Stone & Tukey (1942), which is the basis of the name 'Stone–Tukey theorem'. This paper proves the n-dimensional version of the theorem in a more general setting involving measures. The paper attributes the n = 3 case to Stanislaw Ulam, based on information from a referee; but Beyer & Zardecki (2004) claim that this is incorrect, given Steinhaus's paper, although 'Ulam did make a fundamental contribution in proposing' the Borsuk–Ulam theorem. The two-dimensional variant of the theorem (also known as the pancake theorem) can be proved by an argument which appears in the fair cake-cutting literature (see e.g. Robertson–Webb rotating-knife procedure). For each angle α ∈ [ 0 , 180 ∘ ] {displaystyle alpha in } , we can bisect pancake #1 using a straight line in angle α {displaystyle alpha } (to see this, translate a straight line in angle α {displaystyle alpha } from − ∞ {displaystyle -infty } to ∞ {displaystyle infty } ; the fraction of pancake #1 covered by the line changes continuously from 0 to 1, so by the intermediate value theorem it must be equal to 1/2 somewhere along the way). This means that we can take a straight knife, rotate it at every angle α ∈ [ 0 , 180 ∘ ] {displaystyle alpha in } and translate it appropriately for that particular angle, such that pancake #1 is bisected at each angle and corresponding translation. When the knife is at angle 0, it also cuts pancake #2, but the pieces are probably unequal (if we are lucky and the pieces are equal, we are done). Define the 'positive' side of the knife as the side in which the fraction of pancake #2 is larger. Define p ( α ) {displaystyle p(alpha )} as the fraction of pancake #2 at the positive side of the knife. Initially p ( 0 ) ≥ 1 / 2 {displaystyle p(0)geq 1/2} .