New results on lower bounds for the number of (≤k)(≤k)-facets

In this paper we present two different results dealing with the number of (≤k)(≤k)-facets of a set of points: 1.We give structural properties of sets in the plane that achieve the optimal lower bound 3(k+22) of (≤k)(≤k)-edges for a fixed 0≤k≤⌊n/3⌋−10≤k≤⌊n/3⌋−1; and2.we show that, for k<⌊n/(d+1)⌋k<⌊n/(d+1)⌋, the number of (≤k)(≤k)-facets of a set of nn points in general position in RdRd is at least (d+1)(k+dd), and that this bound is tight in the given range of kk.