Reducing the Large Set Threshold for Oertel's Conjecture on the Mixed-Integer Volume
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914119955775488 |
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| author | Cristi, Andrés Salas, David |
| author_facet | Cristi, Andrés Salas, David |
| contents | In 1960, Grünbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently, in 2014, Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, he proposed that for any convex body $C\subset \mathbb{R}^{n+d}$, there should exist a point $\mathbf{x} \in S=C\cap(\mathbb{Z}^{n}\times\mathbb{R}^d)$ such that for every halfspace $H$ containing $\mathbf{x}$, one has that
\[
\mathcal{H}_d(H\cap S) \geq \frac{1}{2^n}\frac{1}{e}\mathcal{H}_d(S),
\]
where $\mathcal{H}_d$ denotes the $d$-dimensional Hausdorff measure. While the conjecture remains open, Basu and Oertel proved in 2017 that the above inequality holds true for sufficiently large sets, in terms of a measure known as the \emph{lattice width} of a set. In this work, by following a geometric approach, we improve this result by substantially reducing the threshold at which a set can be considered large. We reduce this threshold from an exponential to a polynomial dependency on the dimension, therefore significantly enlarging the family of mixed-integer convex sets over which Oertel's conjecture holds true. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reducing the Large Set Threshold for Oertel's Conjecture on the Mixed-Integer Volume Cristi, Andrés Salas, David Metric Geometry Optimization and Control In 1960, Grünbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently, in 2014, Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, he proposed that for any convex body $C\subset \mathbb{R}^{n+d}$, there should exist a point $\mathbf{x} \in S=C\cap(\mathbb{Z}^{n}\times\mathbb{R}^d)$ such that for every halfspace $H$ containing $\mathbf{x}$, one has that \[ \mathcal{H}_d(H\cap S) \geq \frac{1}{2^n}\frac{1}{e}\mathcal{H}_d(S), \] where $\mathcal{H}_d$ denotes the $d$-dimensional Hausdorff measure. While the conjecture remains open, Basu and Oertel proved in 2017 that the above inequality holds true for sufficiently large sets, in terms of a measure known as the \emph{lattice width} of a set. In this work, by following a geometric approach, we improve this result by substantially reducing the threshold at which a set can be considered large. We reduce this threshold from an exponential to a polynomial dependency on the dimension, therefore significantly enlarging the family of mixed-integer convex sets over which Oertel's conjecture holds true. |
| title | Reducing the Large Set Threshold for Oertel's Conjecture on the Mixed-Integer Volume |
| topic | Metric Geometry Optimization and Control |
| url | https://arxiv.org/abs/2411.11864 |