Ham sandwich theorem by Changqing Li

1. Ham sandwich theoremby Changqing Li Mathematics Discrete geometry Computational geometry Measure theory

2. What is “ham sandwich theorem”? • The volumes of any n-n dimensional solids can always be simultaneously bisected by a (n-1) dimensional hyperplan • In geometry, a hyperplane is a generalization of the two-dimensional plane into an arbitrary number of dimensions. • In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

3. 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. • In two dimensions, the theorem is known as the pancake theorem. • also called the Stone–Tukey theorem

4. 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 by Steinhaus (1938). • Beyer and Zardecki's paper 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. • A more modern reference is Stone & Tukey (1942), which is the basis of the name "Stone–Tukey theorem".

5. 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. • Borsuk–Ulam theorem is that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

6. Measure theory version • In measure theory, Stone & Tukey (1942) 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.

7. 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 points. • 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 a ham sandwich cut for them.

8. Discrete and computational geometry versions • 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.

9. Which field of mathematics does the theorem belong to? • the ham sandwich theorem belongs to measure theory, a branch of mathematics. • Definition of measure theory: A measure is defined as a nonnegative real function from a delta-ring such that

10. Reference • http://mathworld.wolfram.com/HamSandwichTheorem.html • http://mathworld.wolfram.com/Measure.html • http://mathworld.wolfram.com/Hyperplane.html • http://www.guokr.com/article/53059/?page=2