promised last time (lecture 15):
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: (d+1)(r-1)+1 is sharp. Obtaining worse bounds is easy, we proved it for r=4 with (d+2)^2 points from Radon.
Proof: We want to use colorful Caratheodory in a higher dimension.
Before, we switch to cones instead of convex combinations by adding a last 1 coordinate to each point.
Consider the following r mappings, f_1, .., f_r, from R^{d+1} to R^N where N=(d+1)(r-1).
For i<r, "copy" into the i-th block of d coordinates, while f_r= -sum_{i<r} f_i.
Claim: sum_i f_i(u_i)=0 if and only if all u_i are equal.
Define M_i as \cup_j f_j(a_i) for the points, a_i, of our set.
Using colorful Caratheodory, there is a rainbow simplex containing the origin.
That means sum_i l_i f_g(i)(a_i)=0 for some nonnegative l_i numbers for which sum_i l_i=1 and a g: [N+1]->[r] function.
We can define the r partitions as A_j={a_i: g(i)=j}.
Since f_i are all linear, we have 0=sum_i l_i f_g(i)(a_i)=sum_j f_j(sum_{i: a_i\in A_j} l_i a_i).
Using the claim, we have that for all j sum_{i: a_i\in A_j} l_i a_i is the same value, so it is in the cone.
The only problem is if this value is 0, then some A_j might be empty.
But since all last coordiantes are 1, this is impossible as not all l_i are zero.