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

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 Alc\'antara 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 $N^{ 1/3}\log N$.

Categories: math.CO math.NT