The Centerpoint Theorem states that, for any set S of n points in R(d), there exists a point p in R(d) such that every closed halfspace containing p contains at least [n/(d + 1)] points of S. We consider generalizations of the Centerpoint Theorem in which halfspaces are replaced with wedges (cones) of angle alpha. In R(2), we give bounds that are tight for all values of ff and give an O(n) time algorithm to find a point satisfying these bounds. We also give partial results for R(3) and, more generally, R(d).
Discrete Mathematics & Theoretical Computer Science