Corpus ID: 221655547

# Empty axis-parallel boxes

@article{Bukh2020EmptyAB,
title={Empty axis-parallel boxes},
author={B. Bukh and Ting-Wei Chao},
journal={ArXiv},
year={2020},
volume={abs/2009.05820}
}
• Published 2020
• Computer Science, Mathematics
• ArXiv
We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega(d/n)$ as $n\to\infty$ and $d$ is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2\log d/n)$. These improve on the previous best bounds of $\Omega(\log d/n)$ and $O(2^{7d}/n)$ respectively.
1 Citations

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