Algorithms for Center and Tverberg Points
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 inverted right perpendicular n/4 inverted left perpendicular 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 Omega(n(2)) complexity. We then consider sets S of 3n 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.
tverberg point center point arrangements