We follow for the Ham-Sandwich theorem:
http://terrytao.wordpress.com/2008/11/27/the-kakeya-conjecture-and-the-ham-sandwich-theorem/
for polynomial cutting lemma:
http://terrytao.wordpress.com/2010/11/20/the-guth-katz-bound-on-the-erdos-distance-problem/
and for obtaining Szemeredi-Trotter:
http://terrytao.wordpress.com/2011/02/18/the-szemeredi-trotter-theorem-via-the-polynomial-ham-sandwich-theorem/
Borsuk-Ulam theorem:
Let f: S^n \to R^n be a continuous map from the n-dimensional sphere S^n to the Euclidean space R^n which is antipodal
(which means that f(-x)=-f(x) for all x \in S^n). Then f(x)=0 for at least one x \in S^n.
Ex 0: Anyone knows some GOOD real world application?
Other versions exist for primes>2, p points are mapped to 0 if dim S>(p-1)n.
Equivalent versions and consequences:
0, If antipodality is not required, there is a pair of antipodal points with same image.
Ex 1: Prove that this is equivalent to the stated version of BU.
1, To cover S^n, we need n+1 open sets that contain no antipodal points.
2, Brouwer's fixed-point theorem: Any continous function from a ball to itself has a fixed-point.
Ex 2: Prove that this follows (for a disc in the plane) from BU.
3, Ham-Sandwich theorem:
Let U_1,.., U_n be n bounded open sets in R^n.
Then there exists a hyperplane in R^n that divides each of the open sets U_1,...,U_n into two sets of equal volume.
Proof (of HS from BU): Each affine hyperplane is given by an equation <a,x>=b, which gives a point of S^n.
For each check how they divide U_i. This is our function to R^n.
Similarly follows Polynomial Ham-Sandwich theorem:
Let U_1,..., U_{{n+d\choose d}-1} be bounded open sets in R^n.
Then there exists a non-trivial polynomial P: R^n \to R of degree at most d such that
the sets {P>0}, {P<0} partition each of the U_1,...,U_{n+d\choose d}-1} into two sets of equal measure.
Proof is same as before except that now we take polynomials in R^d.
Since there are {n+d\choose d}-1 monoms (choice with repetition!), we are done.
Both of these statements discretize - at most half of each U_i is in {P>0}, {P<0}.
Application: Necklace splitting problem:
Two thieves want to divide necklace with d types of beads among them with at most d cuts.
Solution: Put necklace on moment curve (t,t^2,t^3,..,t^d) and apply Ham-Sandwich.
Polynomial Cutting lemma: If we have m points in the plane, we can partition them with the zero-curve (P=0)
of a degree D polynomial such that each cell has at most O(m/D^2) points.
(It follows from Milnor-Thom theorem that there are at most O(D^2) cells.)
Proof: Repeatedly apply Polynomial Ham-Sandwich.
Cor: If we have at most O(D^2) points, then there is a curve through all of them.
Homework: Suppose we have n points in the plane in general position and a line that goes through one of them.
Rotate the line clockwise around point, until it hits another, then continue rotating around it and so on.
Prove that for any points set there is a starting line that hits each point infinitely many times.