Tverberg Plus Minus.

We prove a Tverberg type theorem: Given a set A⊂Rd<inline-graphic></inline-graphic> in general position with |A|=(r-1)(d+1)+1<inline-graphic></inline-graphic> and k∈{0,1,…,r-1}<inline-graphic></inline-graphic>, there is a partition of A into r sets A1,…,Ar<inline-graphic></inline-graphic> (where |Aj|≤d+1<inline-graphic></inline-graphic> for each j) with the following property. There is a unique z∈⋂j=1raffAj<inline-graphic></inline-graphic> and it can be written as an affine combination of the element in Aj<inline-graphic></inline-graphic>: z=∑x∈Ajα(x)x<inline-graphic></inline-graphic> for every j and exactly k of the coefficients α(x)<inline-graphic></inline-graphic> are negative. The case k=0<inline-graphic></inline-graphic> is Tverberg’s classical theorem. [ABSTRACT FROM AUTHOR]