The integer hull of the set $\{(x,y)\in \mathbb{R}^2: xy\ge N\}$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914311248543744 |
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| author | Balog, Antal Bárány, Imre |
| author_facet | Balog, Antal Bárány, Imre |
| contents | 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 $N^{1/3}\log N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06897 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The integer hull of the set $\{(x,y)\in \mathbb{R}^2: xy\ge N\}$ Balog, Antal Bárány, Imre Combinatorics Number Theory 11H06, 52C05 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 $N^{1/3}\log N$. |
| title | The integer hull of the set $\{(x,y)\in \mathbb{R}^2: xy\ge N\}$ |
| topic | Combinatorics Number Theory 11H06, 52C05 |
| url | https://arxiv.org/abs/2602.06897 |