Centerpoint theorems for wedges

Jeff Erickson (1) , Ferran Hurtado (2) , Pat Morin (3)

1 Department. of Computer Science [Illinois]
2 UPC - Universitat Politècnica de Catalunya = Université polytechnique de Catalogne [Barcelona]
3 Computational Geometry Lab

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).

Domaines: Mathématique discrète [cs.DM]

Jeff Erickson, Ferran Hurtado, Pat Morin. Centerpoint theorems for wedges. Discrete Mathematics and Theoretical Computer Science, 2009, Vol. 11 no. 1 (1), pp.45--54. ⟨10.46298/dmtcs.464⟩. ⟨hal-00988186⟩