New results on lower bounds for the number of -facets

In this paper we present two different results dealing with the number of ( ≤ k ) -facets of a set of points: 1. e structural properties of sets in the plane that achieve the optimal lower bound 3 ( k + 2 2 ) of ( ≤ k ) -edges for a fixed 0 ≤ k ≤ ⌊ n / 3 ⌋ − 1 ; and w that, for k < ⌊ n / ( d + 1 ) ⌋ , the number of ( ≤ k ) -facets of a set of n points in general position in R d is at least ( d + 1 ) ( k + d d ) , and that this bound is tight in the given range of k .