### Centerpoint Theorems for Wedges

*Jeff Erickson, Ferran Hurtado, Pat Morin*

#### Abstract

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 α. In ℝ^{2}, we give bounds that are tight for all values of α 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}.Full Text: PDF PostScript