The integer hull of the set $\{(x,y)\in {\mathbb{R}}^2:xy\ge N\}$

Abstract

The integer convex hull $I(H_N)$ of the set $H_N=\{(x,y)\in {\mathbb{R}}^2:xy\ge N\}$ is the convex hull of the lattice points in $H_N$. The vertices of $I(H_N)$ lie in the square $[1,N{]}^2$. Improving on a recent result of Alcántara et al. ~\cite{Santos} we show that the number of vertices of $I(H_N)$ is of order $N^{1/3}\log N$. We also show that the area of the part of $H_N\setminus I(H_N)$ that lies in the square $[1,N^{2/3}{]}^2$ is also of order $N^{1/3}\log N$.

20 pages, 6 figures