26/02/2022
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Propaganda – p: MachineryIsland Records – 0-96835Vinyl, 12", 45 RPM, ARUS 1985 https://www.discogs.com/Propaganda-p-Machinery/release/135567
The Ham Sandwich Theorem
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Hello. Hi. Welcome to The Ham Sandwich Theorem. 100% dad house. Lotsa mustard. Hold the mayo. Just jamming and discovering music with friends on the interwebs.
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The Ham Sandwich Theorem
The problem of fairly dividing a single object or a group of objects among more than two people (for example, victuals among three cave dwellers, as in Figure 3), even when the object is completely divisible and the values additive, remained unresolved until the 1940s, when the Polish mathematician Hugo Steinhaus made two important discoveries that have inspired much of modern research on fair division. First, he proved in his famous Ham Sandwich Theorem that any three three-dimensional objects (say ham, cheese and bread) may be simultaneously bisected by a single plane (see Figure 4). The objects need not be connected, regular in shape or in any special orientation; more generally, any n objects in n-dimensional space may be simultaneously bisected by a single hyperplane.
In two dimensions, for example, this theorem says that if salt and pepper are sprinkled randomly on a table, a single straight line always exists that will simultaneously separate thesalt into two equal parts and the pepper into two equal parts. On the other hand, there is notalways a line that will simultaneously bisect salt, pepper and sugar on a tabletop (for example,when the salt is all sprinkled tightly around one vertex of a triangle, the pepper around asecond vertex and the sugar around the third). It is crucial that the number of objects not exceed the spatial dimension. So, although Steinhaus's theorem says that a Big Mac may be sliced with a straight (planar) sweep of a knife so thatit simultaneously bisects the bread, meat and cheese, there is no guarantee that any planar bisection also contains equal amounts of the lettuce or "secret sauce." (The example of the Big Mac was chosen to emphasize that even if the bread, meat and cheese are in several pieces, there is still a planar cut that simultaneously bisects these three, or any three, ingredients.)
The Ham Sandwich Theorem suffers one serious additional drawback beyond the dimensional restriction: It is not constructive. That is, although it guarantees that a sandwich bisection exists, it gives no clue as to how to find one.This distinction between nonconstructive and constructive proofs is an important one in mathematics, especially in applications. A nonconstructive existence proof argues indirectly, as in "if the temperature at noon yesterday was 20 degrees and it is now 30 degrees, then sometime in between it must have been exactly 25 degrees." The argument gives no indication of when it was 25 degrees.
A constructive proof, on the other hand, argues existence by providing an algorithm or procedure for finding the objectin question, and this is exactly what Steinhaus's second major contribution did. It proves the existence of fair divisions by giving a practical and general procedure for dividing an inhomogeneous irregular object such as fruitcake among anarbitrary number of people so that each receives a portion he considers a fair share. Each of n people will receive a piecehe values at least one-nth of the cake, even though different individuals may have different values—one preferring the frosting, another the nuts and so forth. Since Steinhaus's discovery, a number of similar algorithms have been devised based on ideas of rotating reduction, iterated cut-and-choose and elegant sliding-knife techniques (see Figure 5).
