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) e structural properties of sets in the plane that achieve the optimal lower bound 3 ( k + 2 2 ) of (⩽ k)-edges for a fixed k ⩽ ⌊ n / 3 ⌋ − 1 ; w that the new lower bound 3 ( k + 2 2 ) + 3 ( k − ⌊ n 3 ⌋ + 2 2 ) 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 ⩽ ⌊ 5 n / 12 ⌋ − 1 ; w that for k < n / 4 the number of (⩽ k)-facets of a set of n points in R 3 in general position is at least 4 ( k + 3 3 ) , and that this bound is tight in that range.