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

TL;DR

This paper investigates the structure of the integer convex hull of the set defined by the inequality xy ≥ N, revealing that the number of vertices and the area of the non-included region grow roughly as N^{1/3} log N, refining previous bounds.

Contribution

It establishes new bounds on the number of vertices and the area of the difference between the set and its integer convex hull, improving recent results by Alcántara et al.

Findings

Number of vertices of the integer hull is of order N^{1/3} log N.

Area of the non-included region in a specific square is of order N^{1/3} log N.

Provides tighter asymptotic bounds on the structure of the integer hull.

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 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 $N^{1/3}\log N$.