Empty Monochromatic Simplices

Abstract

Let S be a k-colored (nite) set of n points in R d , d 3, in general position, that is, no (d+1) points of S lie in a common (d 1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3 k d we provide a lower bound of ( n d k+1+2 d ) and strengthen this to ( n d 2=3 ) for k = 2. On the way we provide various results on triangulations of point sets in R d . In particular, for any constant dimension d 3, we prove that every set of n points (n suciently large), in general position in R d , admits a triangulation with at least dn + (log n) simplices.