Кез келген үш денені бір жазықтықпен тең бөлу теоремасы

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 even possible if the objects overlap. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without taking the trouble to 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 1=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 , bread and cheese and ham , or bread and butter and ham 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 .

According to , the earliest known paper about the ham sandwich theorem, specifically the 1=n = 3 case of bisecting three solids with a plane, is a 1938 note in a Polish mathematics journal 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. A more modern reference is , 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 1=n = 3 case to Stanislaw Ulam, based on information from a referee; but claim that this is incorrect, given the note mentioned above, 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 , a straight line ("knife") of angle can bisect pancake #1. To see this, translate [move parallelly] a straight line of angle from to ; 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. It is possible that an entire range of translations of our line yield a fraction of 1/2; in this case, it is a canonical choice to pick the middle one of all such translations. 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. We now turn the knife, and translate it as described above. When the angle is , define as the fraction of pancake #2 at the positive side of the knife. Initially The function is continuous, since small changes in the angle lead to small changes in the position of the knife. When the knife is at angle 180, the knife is upside down, so By the intermediate value theorem, there must be an angle in which Cutting at that angle bisects both pancakes simultaneously.

In measure theory, proved two more general forms of the ham sandwich theorem. Both versions concern the bisection of n subsets X1, X2, , Xn of a common set X, where X has a Carathéodory outer measure and each Xi has finite outer measure. Their first general formulation is as follows: for any continuous real function , there is a point p of the n sphere S^(n) and a real number s0 such that the surface 1=f(p,x) = s0 divides X into f(p,x) < s0 and f(p,x) > s0 of equal measure and simultaneously bisects the outer measure of X1, X2, , Xn. The proof is again a reduction to the Borsuk Ulam theorem. This theorem generalizes the standard ham sandwich theorem by letting 1=f(s,x) = s1x1 + + snxn. Their second formulation is as follows: for any n + 1 measurable functions f0, f1, , fn over X that are linearly independent over any subset of X of positive measure, there is a linear combination 1=f = a0f0 + a1f1 + + anfn such that the surface 1=f(x) = 0, dividing X into f(x) < 0 and f(x) > 0, simultaneously bisects the outer measure of X1, X2, , Xn. This theorem generalizes the standard ham sandwich theorem by letting 1=f0(x) = 1 and letting fi(x), for i > 0, be the i th coordinate of x.

The original theorem works for at most n collections, where n is the number of dimensions. To bisect a larger number of collections without going to higher dimensions, one can use, instead of a hyperplane, an algebraic surface of degree k, i. e., an (n−1)–dimensional surface defined by a polynomial function of degree k:

Given measures in an n–dimensional space, there exists an algebraic surface of degree k which bisects them all. This generalization is proved by mapping the n–dimensional plane into a dimensional plane, and then applying the original theorem. For example, for 1=n = 2 and 1=k = 2, the 2–dimensional plane is mapped to a 5–dimensional plane via:

(x, y) → (x, y, x^(2), y^(2), xy).