Centerpoint theorems for wedges

Genera

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 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