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

## 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## Preview

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