Let C be the convex-hull of a set of points S with integral coordinates in the plane. It is well-known that |C| ≤ cD2/3 for some constant c where D is the diameter of S: i.e. the maximum distance between any pair of points in S. It has been shown that c = 7.559.. for an arbitrary S, and c = 3.496.. in the special case when S is a ball centered at the origin in the plane. In this paper we show that c = 12/ 3v 4p2 = 3.524.. is sufficient for an arbitrary set of lattice points S of diameter D in the plane, and |C| ~ 12 3v2/(9p2) D2/3 = 3.388..D2/3 is achieved asymptotically. Our proof is based on the construction of a special set in ?rst quadrant, and the analysis of the result involves the calculation of the average order of certain number-theoretical functions associated with the Euler totient function φ(n).
Convex Hull Lattice Point Integer Point Average Order Convex Lattice
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.