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

TL;DR

This work determines the precise growth rate of the vertex count of the integer hull $I(H_N)$, where $H_N=\{(x,y): xy\ge N, x,y>0\}$. By combining a Minkowski-type bound that forces vertices into a narrow hyperbola strip with divisor-sum counts for lattice points on hyperbolas, the authors prove $f_0(I(H_N))$ is asymptotically of order $N^{1/3}\log N$, matching a lower and an upper bound. They also bound the area of the region $H_N\setminus I(H_N)$ inside $[1,N^{2/3}]^2$ by the same order, using curvature and cap-area analyses together with flatness considerations. The results connect lattice polytope geometry with the theory of random polytopes, floating bodies, and curvature-based area estimates, providing a refined understanding of integer hulls for hyperbola-like convex sets with potential implications for integer programming and the geometry of numbers.

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

Figures (6)

• Figure 1: $L$ intersecting $H^1$ and $H^2$.
• Figure 2: $S^*$ and $S^1$.
• Figure 3: $v_1,v_2\in V$ and a few $z \in NV$ with $z\to v_i$.
• Figure 4: $H^1$ and the osculating circle.
• Figure 5: The cap $C^p$ and the osculating circle.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Claim 5.1
• Claim 6.1