There is a new thread for polymath4, the url is:
Archive for October, 2009
There is a site for answering questions related to mathematics. The url is here:
As I understand it the question should be of interest to at least one mathematician. People are rated in regards to their responses and with higher ratings come higher privileges. There is more information at the site.
There is a article on polymath here:
Let us generalize this problem to r sets and dimension d:
We have r(d+1) -1 points on the plane. We want to show that we can divide them into r-1 sets of d+1 points and one set of d-2 such that the intersection of the convex hulls of all r sets is not empty.
We take (r-1)*(d+1) +1 of the points and apply Tverberg’s theorem.
It gives us r disjoint nonempty sets which together contain all (r-1)*(d+1) +1 points. The intersection of the convex hulls of these sets is nonempty. Let one of the points in the intersection of the convex hulls be x.
If any of the r sets are bigger than d+1 then by Caratheodory’s theorem there is a d+1 subset that contains x. We take that as the new set in place of the old set and add the additional points to sets less than d+1 making sure none of the additions causes a set to be greater than d+1. We repeat this process untill all of the sets are three or less. Then we add the remaining d-1 points to a sets with d points or less again making sure no set goes over d+1. The result is (r)*(d+1) +1 points divided into r sets of d+1 or less. The only way to do this is r-1 sets of d+1 and one set of d points which gives the desired theorem.
We can add more points and keep the distribution even since adding a point to a set of points increases the convex hull so the intersection of convex hulls still contains x. So we have the following if we have r(d+1) -1 points or more we can divide them into r nonempty sets, the size of each set differing by at most one element from each other set such that the intersection of the convex hulls of the sets is nonempty.
There is a new thread for Polymath3:
Some of the things I have mentioned previously have been expanded in various ways. Let me list them in this post.
First the thread about mathematicians and blogging is now an article it is available here:
Here is a thread about the first draft:
Also the conversation about complexity lower bounds will be about 7 parts, three are posted so far.
All seven will eventually be at: