COCOON 2001: Computing and Combinatorics pp 111-120

# An Improved Upper Bound on the Size of Planar Convex-Hulls

• Abdullah N. Arslan
• Ömer Eğecioğlu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2108)

## Abstract

Let C be the convex-hull of a set of points S with integral coordinates in the plane. It is well-known that |C| ≤ cD 2/3 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) D 2/3 = 3.388..D 2/3 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).

## Keywords

Convex Hull Lattice Point Integer Point Average Order Convex Lattice
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