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NAMING 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 each be 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 Pancake s 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. The ham sandwich theorem has no relationship to the " Squeeze Theorem " (sometimes called the "sandwich theorem"). HISTORY According to Beyer and Zardecki (2004), the earliest known paper about the ham sandwich theorem, specifically the ''d'' = 3 case of bisecting three solids with a plane, is by Steinhaus and others (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 and 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 Stanisław Marcin 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 . REDUCTION TO THE BORSUK-ULAM THEOREM The ham sandwich theorem can be proved as follows using the Borsuk-Ulam Theorem . This proof follows the one described by Steinhaus and others (1938), attributed there to Stefan Banach , for the ''n'' = 3 case. Let ''A''1, ''A''2, ..., ''A''''n'' denote the ''n'' objects that we wish to simultaneously bisect. Let ''S'' be the Unit (''n'' − 1)- Sphere in , centered at the Origin . For each point ''p'' on the surface of the sphere ''S'', we can define a Continuum of oriented Hyperplane s perpendicular to the ( Normal ) Vector from the origin to ''p'', with the "positive side" of each hyperplane defined as the side pointed to by that vector. By the Intermediate Value Theorem , every family of such hyperplanes contains at least one hyperplane that bisects the bounded object ''A''''n'': at one extreme translation, no volume of ''A''''n'' is on the positive side, and at the other extreme translation, all of ''A''''n'''s volume is on the positive side, so in between there must be a translation that has half of ''A''''n'''s volume on the positive side. If there is more than one such hyperplane in the family, we can pick one canonically by choosing the midpoint of the interval of translations for which ''A''''n'' is bisected. Thus we obtain, for each point ''p'' on the sphere ''S'', a hyperplane π(''p'') that is perpendicular to the vector from the origin to ''p'' and that bisects ''A''''n''. Now we define a function ''f'' from the (''n'' − 1)-sphere ''S'' to (''n'' − 1)-dimensional Euclidean Space as follows: f ::volume of ''A''1 on the positive side of π(''p''), ::volume of ''A''2 on the positive side of π(''p''), ::..., ::volume of ''A''''n''−1 on the positive side of π(''p'') :). This function ''f'' is Continuous . By the Borsuk-Ulam Theorem , there are Antipodal Points ''p'' and ''q'' on the sphere ''S'' such that ''f''(''p'') = ''f''(''q''). Antipodal points ''p'' and ''q'' correspond to hyperplanes π(''p'') and π(''q'') that are equal except that they have opposite positive sides. Thus, ''f''(''p'') = ''f''(''q'') means that the volume of ''A''''i'' is the same on the positive and negative side of π(''p'') (or π(''q'')), for ''i'' = 1, 2, ..., ''n'' − 1. Thus, π(''p'') (or π(''q'')) is the desired ham sandwich cut that simultaneously bisects the volumes of ''A''1, ''A''2, ..., ''A''''n''. MEASURE THEORETIC STATEMENT In measure theory, a more general form of the ham sandwich theorem is due to Stone and Tukey (1942): for any ''n'' , that simultaneously divides each of the ''n'' subsets in half with respect to the measure. DISCRETE AND COMPUTATIONAL GEOMETRY VERSIONS In Discrete Geometry and Computational Geometry , the ham sandwich theorem usually refers to the special case in which each of the sets being divided is a Finite set of Point s. Here the relevant measure is the Counting Measure , which simply counts the number of points on either side of the hyperplane. In two dimensions, the theorem can be stated as follows: :For a finite set of points in the plane, each colored "red" or "blue", there is a Line that simultaneously bisects the red points and bisects the blue points, that is, the number of red points on either side of the line is equal and the number of blue points on either side of the line is equal. There is an exceptional case when a point lies on the line. In this situation, we count the point as being on either both sides of the line or on neither side of line. This exceptional case is actually required for the theorem to hold, in case the number of red points or the number of blue is odd, and still each set must be bisected. In computational geometry, this ham sandwich theorem leads to a computational problem, the ham sandwich problem. In two dimensions, the problem is this: given a finite set of ''n'' points in the plane, each colored "red" or "blue", find their ham sandwich cut. After a series of papers with various solutions to this problem, Lo and Steiger (1990) found an optimal '' O ''(''n'')-time Algorithm . This algorithm is Randomized , but Lo, Matousek, and Steiger (1994) showed how it could be derandomized into a Deterministic Algorithm with ''O''(''n'') running time in the worst case. "LEFTOVERS" Byrnes, Cairns and Jessup (2001) showed that it is not always possible to position the hyperplane correctly just by cutting through the objects' Center Of Gravity . REFERENCES
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