Proc. 7th Annual International Computing and Combinatorics Conference
(COCOON'01), LNCS 2108, Guilin, China, August 2001, pp. 111-120.
Abdullah N. Arslan and Ömer Egecioglu
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 $|C| \leq c D^{2/3}$
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 $c= 12/{\sqrt[3]{4 \pi^2}}= 3.524..$ is sufficient
for an arbitrary set of lattice points
S of diameter D in the plane, and
$|C| \sim 12 \sqrt[3]{2/(9 \pi^2)}D^{2/3}
= 3.388..D^{2/3}$ is achieved asymptotically.
Our proof is based on the construction of a special set
in first 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 $ \phi (n)$.
omer@cs.ucsb.edu