4. Sarkaria’s proof:
Tverberg’s theorem (1965): Let be points in , . Then there is a partition of such that .
Proof: We can assume that . First suppose that the points belong to the -dimensional affine space in of points with the property that the sum of all coordinates is 1. Next consider another -dimensional vector space and vectors in , such that is the only linear relation among the s. (So we can take as the standard basis in and . )
Now we consider the tensor product .
Nothing to be scared of: can be regarded just as the -dimensional vector space of d+1 by matrices. We can define the tensor product of two vectors and , as the (rank-one) matrix .
Consider in U the following sets of points:
Note that 0 is in the convex hull of every . Indeed it is the sum of the elements in . (And thus . )
By the Colorful Caratheodory Theorem we can Continue reading