Centerpoint theorems for wedges

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 a. In M ², we give bounds that are tight for all values of a 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.