### Ham sandwich theorem by Changqing Li

```Ham sandwich theorem
by Changqing Li
Mathematics
•Discrete geometry
•Computational geometry
•Measure theory
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 twodimensional 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.
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
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".
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.
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.
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.
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.
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
Reference
• http://mathworld.wolfram.com/HamSandwic
hTheorem.html
• http://mathworld.wolfram.com/Measure.html
• http://mathworld.wolfram.com/Hyperplane.h
tml
• http://www.guokr.com/article/53059/?page=
2
```