Splitting a Configuration in a Simplex

This paper presents a new method of partition, namedπ-splitting, of a point set ind-dimensional space. Given a pointG in ad-dimensional simplexT, T(G;i) is the subsimplex spanned by G and the ith facet ofT. LetS be a set ofn points inT, and letπ be a sequence of nonnegative integers π1, ..., nd+1 satisfying σi=1d+1π1=n Theπ-splitter of (T, S) is a pointG inT such thatT(G;i) contains at leastπi points ofS in its closure for everyi=1, 2, ...,d + 1. The associated dissection is the re-splitting.