Algorithms for center and Tverberg points

Abstract

Given a set S of n points in R 3 , a point x in R 3 is called center point of S if every closed halfspace whose bounding hyperplane passes through x contains at least ⌈ n /4⌉ points from S . We present a near-quadratic algorithm for computing the center region , that is the set of all center points, of a set of n points in R 3 . This is nearly tight in the worst case since the center region can have Ω( n 2 ) complexity. We then consider sets S of 3 n points in the plane which are the union of three disjoint sets consisting respectively of n red, n blue, and n green points. A point x in R 2 is called a colored Tverberg point of S if there is a partition of S into n triples with one point of each color, so that x lies in all triangles spanned by these triples. We present a first polynomial-time algorithm for recognizing whether a given point is a colored Tverberg point of such a 3-colored set S .