In the third of this series of posts I want to present Tverberg’s theorem. It is a generalization of Radon’s theorem: If m is greater than (r-1)*(d+1) +1 then any m points in d dimensional space can be divided into r sets that are not empty such that the intersection of the convex hulls of the sets is not empty.

Also let me introduce Caratheodory’s theorem if in d dimensional space x is in the convex hull of n points and n is greater than d+1 there is a subset of d+1 of the n points such that the convex hull of the d+1 points contains x. More information about these theorems is here:

http://gilkalai.wordpress.com/2008/11/24/sarkarias-proof-of-tverbergs-theorem-1//

In the next of these series of posts I want to start using these tools on the original problem.

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