Ham sandwich theorem
theorem that any three objects in space can be simultaneously bisected by a plane
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 each one of them in half (with respect to their measure, e.g. volume) with a single (n − 1)-dimensional hyperplane. This is possible even if the objects overlap.
It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without stating the theorem in the n-dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey.
Naming
The ham sandwich theorem takes its name from the case when n = 3 and the three objects to be bisected are the ingredients of a ham sandwich. Sources differ on whether these three ingredients are two slices of bread and a piece of ham (Peters 1981), bread and cheese and ham (Cairns 1963), or bread and butter and ham (Dubins & Spanier 1961). In two dimensions, the theorem is known as the pancake theorem to refer to the flat nature of the two objects to be bisected by a line (Cairns 1963).
History
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 a 1938 note in a Polish mathematics journal (Editors 1938). Beyer and Zardecki's paper includes a translation of this note, which 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 note 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?" The note then offers a proof of the theorem.
Begin with the source’s own compact description: “Ham sandwich theorem” is theorem that any three objects in space can be simultaneously bisected by a plane. The dossier treats that line as a proposition to test through sandwich, theorem and three, not as a finished interpretation.
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This entry incorporates text from “Ham sandwich theorem” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.