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.The existence of aπ-splitting is shown for any (T, S) andπ, and two efficient algorithms for finding such a splitting are given. One runs inO(d2n logn + d3n) time, and the other runs inO(n) time if the dimensiond can be considered as a constant. Applications of re-splitting to mesh generation, polygonal-tour generation, and a combinatorial assignment problem are given.