New results on lower bounds for the number of (⩽ k)-facets: (extended abstract)

In this paper we present three different results dealing with the number of (⩽ k)-facets of a set of points:(i)We give structural properties of sets in the plane that achieve the optimal lower bound of (⩽ k)-edges for a fixed k⩽⌊n/3⌋−1;(ii)We show that the new lower bound for the number of (⩽ k)-edges of a planar point set shown in [O. Aichholzer, J. García, D. Orden, and P. A. Ramos. New lower bounds for the number of (⩽ k)-edges and the rectilinear crossing number of K. Discrete and Computational Geometry, in press] is optimal in the range ⌊n/3⌋⩽k⩽⌊5n/12⌋−1;(iii)We show that for k