Points as obstacles
For n points in general position, how many points are needed to block visibility?
In the worst case asymptotically n^2, as any two segments meet just once.
In the best case asymptotically at least n, as shown by convex position.
Denote this number by b(n).
Ex 0: Prove that b(n)>=2n-3.
Thm [Pach, based on Behrend's construction]: b(n)<= n e^O(sqrt{log n}).
Proof: In fact, we will block the midpoints of each segment.
First take a grid in m-dim, [s]^m, where m=sqrt{log n} and s will be about e^O(sqrt{log n}).
By the pigeonhole principle, there is a sphere of radius r^2<ms^2 that contains at least s^{m-2}/m points.
We select s to be the smallest integer such that s^{m-2}>=mn, so s< n^{2/(m-2)}< e^O(sqrt{log n}).
The blocking points are the points of halfgrid, (2s)^m points.
So b(n)<= mn 2^m s^2<= n e^O(sqrt{log n}).
If the points are in convex position, then define minimum as cb(n).
Thm: cb(n) =\Omega(nlogn).
Hw: Prove the theorem.
Generalization of intersection theorems for convex sets
Fractional Helly theorem: For all d and 0<a<1, there is a 0<b, such that
if we have n>d convex sets in d-dim such that
at least an a fraction of all possible d+1-tuples of the convex sets, that is >a{n \choose d+1} intersect,
then >bn of the convex sets intersect.
Note: If a=1, then b=1, this is Helly.
We prove b>=a/(d+1). Best possible is 1-(1-a)^{1/(d+1)}.
Ex 1: Compute a and b for the following system: n-m+d hyperplanes in general position and m-d very big sets.
Proof: We can suppose sets are compact by taking them to be the convex hulls of their relevant points.
Compact sets have unique lex min point.
Lemma: If v is lex min point of intersection of some convex sets,
then it is also lex min point of intersection of at most d of these convex sets.
Proof: Apply Helly for the convex sets and the set C={x<v}.
If indirectly any d intersects C, v was not lex min point.
Proof of frac Helly follows by taking all d-tuples and their lex min points.
One must be lex min for at least a{n \choose d+1} / (n\choose d} = a(n-d)/(d+1) of the d+1 tuples.
This means d+a(n-d)/(d+1)>an/(d+1) convex sets, we are done.
Hw: Colorful Helly theorem:
d+1 family of convex sets in d-dim such that taking one set from each family always gives a non-empty intersection.
Prove that there is a family whose sets have a common point.
Hint: This point is the max of the lex min points of the colorful d-tuples.
Colorful Caratheodory theorem: Suppose we have d+1 sets in d-dim whose convex hull all contain the origin.
Then we can select one point from each such that the convex hull of these points also contains the origin.
Proof: Using Caratheodory we can suppose that each convex set has at most d+1 points.
Take the colorful d+1 tuple closest to the origin.
If it does not contain it, take its face containing its closest point to origin.
We can go closer by replacing a point not on this face with another from the same set.
Ex 2: Very hard! (Schweitzer 2011: http://mathproblems123.wordpress.com/2012/03/06/miklos-schweitzer-2011/)
Let C_1,...,C_d be compact and connected sets in R^d, and suppose that the convex hull of any C_i contains the origin.
Prove that for all i there exists c_i from C_i such that the convex hull of {c_1,c_2,..,c_d} contains the origin.
For d=2 it is not hard.
Tverberg theorem: Given (d+1)(r-1)+1 points in d-dim, we can partition them into r sets such that their convex hulls intersect.
Note: If r=1, this is Radon's lemma. (d+1)(r-1)+1 is sharp. Obtaining worse bounds is easy.
Ex 3: Tverberg implies Centerpoint theorem.
Proof: To come next class.
Colorful Tverberg theorem: (without proof)
For every r,d there is a t such that given d+1 sets of t points, we can select r transversals whose intersection is non-empty.
Note: Midterm 5 asked to prove t(d,2)=2.
It is known that t(d,r)<4r and conjectured to be r, which is verified in special cases.