Archivo Digital UPM: No conditions. Results ordered -Date Deposited. 2019-12-08T07:09:12ZEPrintshttp://oa.upm.es/style/images/logo-archivo-digital.pnghttp://oa.upm.es/2010-10-05T10:03:33Z2016-04-20T13:40:36Zhttp://oa.upm.es/id/eprint/4455This item is in the repository with the URL: http://oa.upm.es/id/eprint/44552010-10-05T10:03:33ZMore about lower bounds for the number of k-facetsIn this paper we present two results dealing with the number of (• k)-facets of a set of points:
•² In R2, we use the notion of ²-net to give structural properties of sets that achieve the optimal lower bound 3¡k+22 ¢ of (• k)-edges for a ¯xed 0 • k • bn=3c ¡ 1;²
•In Rd, we show that for k < bn=(d + 1)c the number of (• k)-facets of a set of n points in general position is at least (d + 1)¡k+dd ¢, and that this bound is tight in that range.Oswin AichholzerJesus Garcia Lopez de LacalleDavid Orden MartínPedro Ramos2010-04-30T09:31:14Z2016-04-20T12:34:41Zhttp://oa.upm.es/id/eprint/2969This item is in the repository with the URL: http://oa.upm.es/id/eprint/29692010-04-30T09:31:14ZNew results on lower bounds for the number of k-facetsIn 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 3_k+2 2 _ of (≤ k)-edges for a fixed k ≤ [n/3 ]− 1; (ii) We show 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 is optimal in the range [n/3] ≤ k ≤ [5n/12] − 1; (iii) We show that for k < n/4 the number of (≤ k)-facets of a set of n points in R3 in general position is at least 4((k+3 )3 ), and that this bound is tight in that range.Oswin AichholzerJesus Garcia Lopez de LacalleDavid Orden MartínPedro A. Ramos Alonso